1100 lines
38 KiB
TeX
1100 lines
38 KiB
TeX
\documentclass[11pt]{amsart}
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\usepackage[margin=1in]{geometry}
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\usepackage{amsmath,amssymb,amsthm,mathtools}
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\usepackage{microtype}
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\usepackage{booktabs}
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\usepackage{enumitem}
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\usepackage{xcolor}
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\usepackage{hyperref}
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\usepackage[nameinlink,capitalize,noabbrev]{cleveref}
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\newtheorem{theorem}{Theorem}[section]
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\newtheorem{proposition}[theorem]{Proposition}
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\newtheorem{corollary}[theorem]{Corollary}
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\newtheorem{lemma}[theorem]{Lemma}
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\theoremstyle{definition}
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\newtheorem{definition}[theorem]{Definition}
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\newtheorem{remark}[theorem]{Remark}
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\newcommand{\cnum}{c}
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\newcommand{\dist}{\operatorname{dist}}
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\newcommand{\diam}{\operatorname{diam}}
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\newcommand{\eps}{\varepsilon}
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\newcommand{\whp}{\text{with high probability}}
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\newcommand{\Occ}{\operatorname{Occ}}
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\newcommand{\iotae}{\iota}
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\hypersetup{
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colorlinks=true,
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linkcolor=blue!50!black,
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citecolor=blue!50!black,
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urlcolor=blue!50!black,
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pdftitle={Ball-Occupation Certificates under Coarse Graph Projections},
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pdfauthor={Levi Neuwirth},
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pdfkeywords={Cops and Robber, Meyniel's conjecture, graph products, degree reduction, expansion}
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}
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\title[Ball-Occupation Certificates under Coarse Graph Projections]{Ball-Occupation Certificates under Coarse Graph Projections\\
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Degree Reduction, Square-Root Hard Families, and Toroidal Barriers}
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\author{Levi Neuwirth}
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\address{Brown University}
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\date{July 2026}
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\subjclass[2020]{05C57, 05C40, 05C12}
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\keywords{Cops and Robber; Meyniel's conjecture; graph products; degree reduction; coarse graph projections; expansion; occupation certificates}
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\begin{document}
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\begin{abstract}
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We isolate an abstract strategy-transfer principle for Cops and Robber. Let
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$\pi:V(H)\to V(G)$ have fibers of order at most $P$, and suppose that for a
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scale factor $\lambda\ge1$ the distances between distinct fibers satisfy
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\[
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\lambda(\dist_G(u,v)-1)+1
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\le \dist_H(x,y)
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\le \lambda(\dist_G(u,v)+2)-2,
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\]
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while every fiber has diameter at most $2(\lambda-1)$. If the $N$-vertex
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base graph has uniform multi-source growth at the square-root scale, then a
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Hall assignment occupies a lifted macro-ball before the robber can leave it,
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giving
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\[
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\cnum(H)\le C P\bigl(d^3+\log(ePN)\bigr)\sqrt N
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\]
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and capture time at most $\lambda R$. The theorem uses only the displayed
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projection properties, not the internal form of the fibers.
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Iterated degree reduction of Hosseini--Mohar--Gonzalez Hermosillo de la Maza
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has exactly this geometry, with $\lambda=3^k$. Applied to
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$G(N,p)$ of expected degree $(\log N)^4$, it yields connected subcubic graphs
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of order $M$ with
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\[
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h(H)\ge (\log M)^{-O(1)},
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\qquad
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M^{\frac12-\frac{9+o(1)}{2\log\log\log M}}
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\le \cnum(H)
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\le \sqrt M\,(\log M)^{O(1)}.
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\]
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Thus the known stressing family has the square-root exponent, with a
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polylogarithmically tight upper bound; the lower convergence to $1/2$ is only
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triple-logarithmic.
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Finally, we prove a sharp limitation of the strategy class. If a common
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prepositioned bank must, after learning a robber start $v$, match distinct cops
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to every vertex of $B(v,R)$ using travel distance at most $R$, then
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\[
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\Occ_R(G)
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\ge
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\frac{\sum_v |B(v,R)|}{\max_x |B(x,2R)|}.
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\]
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For every fixed $k\ge2$, the Cartesian torus $C_L^{\square k}$ has
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$|V|=L^k$, vertex expansion $\Theta_k(L^{-1})$, exact cop number $k+1$, and
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$\Occ_R=\Omega_k(L^k)$ for every radius $R$. Hence, for every $\delta>0$,
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there are bounded-degree graphs with $h(G)\ge |G|^{-\delta}$ and constant cop
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number on which one-shot occupation still costs a linear number of cops. A
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cubic four-cycle replacement gives a degree-three instance at exponent
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$\delta=1/2$. Any universal robustness theorem must therefore use adaptive
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multi-round pursuit, cop reuse, or another strategy not reducible to one-shot
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occupation.
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\end{abstract}
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\maketitle
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\section{Introduction and main conclusions}
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The multi-cop version of Cops and Robber was developed by Aigner and Fromme,
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who proved that three cops suffice on every planar graph
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\cite{AignerFromme}. Meyniel's conjecture asks whether every connected
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$n$-vertex graph has cop number $O(\sqrt n)$. The current best universal
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upper bound remains
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\[
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\frac{n}{2^{(1-o(1))\sqrt{\log_2 n}}},
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\]
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proved independently by Lu--Peng and Scott--Sudakov
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\cite{LuPeng,ScottSudakov}. Bose--Esperet--Hodor--Joret--Micek--Rambaud
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recently extended the same scale of bound from graph order to vertex-cover
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number \cite{CurrentFrontier}. Expansion is one of the principal settings in
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which polynomial savings are known: Bradshaw--Hosseini--Mohar--Stacho obtain
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weak Meyniel bounds from bounded-degree expansion restricted to sublinear set
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scales \cite{BHMS}, while Clow's withdrawn preprint developed a closely
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related structural program connecting failure of weak Meyniel to high-cop
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expanding examples \cite{Clow}.
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The motivation here is the effect of bounded-degree replacement gadgets on
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pursuit. The degree-reduction construction of Hosseini--Mohar--Gonzalez
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Hermosillo de la Maza (HMGHM) preserves lower bounds on cop number and
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produces subcubic graphs with cop number $M^{1/2-o(1)}$ \cite{HMGHM}. A
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natural converse question is whether a useful upper strategy on the base
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graph survives the replacement tower.
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An arbitrary winning strategy does not lift transparently. Moving one step
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in the quotient may require a squad dispersed through a cloud to reorganize
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while the robber continues moving. The successful object is narrower and
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more stable: an \emph{occupation certificate} that assigns distinct cops to
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all vertices of a region before the robber can leave it. Distance stretching
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slows both deployment and escape, and a bounded normalized additive error
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leaves a strict timing margin.
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The first result is therefore stated for an abstract projection, rather than
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for the HMGHM gadget. The gadget enters only later, through an exact metric
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calculation. The resulting upper bound is stronger quantitatively than the
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notation $M^{1/2+o(1)}$ suggests: it is $\sqrt M$ times a polylogarithmic
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factor. By contrast, the available lower bound approaches the square-root
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exponent at the triple-logarithmic rate displayed in the abstract. These two
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facts should not be conflated merely because both can be written
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$M^{1/2+o(1)}$.
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The final result marks the boundary of the mechanism. A one-shot occupation
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certificate needs polynomial ball amplification between radii $R$ and $2R$.
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Polynomially weak expansion alone does not supply this. For every fixed
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$k$, the Cartesian tori $C_L^{\square k}$ have bounded metric doubling, exact
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cop number $k+1$, and linear one-shot occupation cost at every radius. Taking
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$k>1/\delta$ puts these examples inside every window
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$h(G)\ge |G|^{-\delta}$. A separate cubic replacement retains the barrier at
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$\delta=1/2$. The obstacle in the universal problem is therefore not degree
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reduction itself; it is the need for adaptive reuse over many weak-growth
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layers.
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\section{Scale-adaptive cores and the robustness window}
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For a connected graph $J$, write
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\[
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h(J)=\min_{\varnothing\ne A\subseteq V(J),\ |A|\le |J|/2}
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\frac{|\partial_J A|}{|A|}.
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\]
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The following elementary reduction explains why polynomially weak expansion
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is the relevant robustness window for the universal problem.
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\begin{proposition}[Scale-adaptive induced core]\label{prop:adaptive-core}
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Let $J$ be a connected graph of order $n$, and let
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$\eta(1)\ge\cdots\ge\eta(n)\ge0$. Then $J$ contains a connected induced
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subgraph $K$, of order $m$, such that
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\[
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\boxed{h(K)\ge\eta(m)}
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\qquad\text{and}\qquad
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\boxed{\cnum(J)\le \cnum(K)+\sum_{j=m+1}^{n}\eta(j).}
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\]
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\end{proposition}
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\begin{proof}
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Maintain the connected induced region $J_i$ containing the robber. Whenever
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$|J_i|=m_i$ and $h(J_i)<\eta(m_i)$, choose
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$A_i\subseteq V(J_i)$ with $0<|A_i|\le m_i/2$ and
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$|\partial_{J_i}A_i|<\eta(m_i)|A_i|$, and occupy its boundary. The robber is
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then confined to one component $J_{i+1}$ of
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$J_i-\partial_{J_i}A_i$. Whether that component lies inside $A_i$ or outside
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it, one has
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\[
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|A_i|\le m_i-m_{i+1}.
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\]
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Consequently the separator costs at most
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\[
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\eta(m_i)(m_i-m_{i+1})
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\le
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\sum_{j=m_{i+1}+1}^{m_i}\eta(j).
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\]
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These integer intervals are disjoint along the robber's nested component
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chain. The process terminates at a connected induced $K$ with
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$h(K)\ge\eta(|K|)$, and the separator costs telescope to the displayed sum.
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\end{proof}
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Taking $\eta(j)=j^{-a}$ shows that a polynomial cop-number saving on
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subcubic graphs with $h(K)\ge |K|^{-a}$ would imply a weak form of Meyniel for
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arbitrary graphs after the bounded-degree transfer of
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Hosseini--Mohar--Gonzalez Hermosillo de la Maza
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\cite[Corollary~8]{HMGHM}. Bradshaw--Hosseini--Mohar--Stacho already treat
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constant expansion restricted to sublinear set scales \cite{BHMS}; the
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unresolved axis in this reduction is expansion that itself shrinks
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polynomially.
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\section{Coarse occupation projections}
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\begin{definition}[Coarse occupation projection]\label{def:projection}
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Let $G$ and $H$ be connected graphs. A surjection
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$\pi:V(H)\to V(G)$ is a $(\lambda,P)$-occupation projection if, writing
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$F_v=\pi^{-1}(v)$,
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\begin{enumerate}[label=\textup{(\roman*)}]
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\item $|F_v|\le P$ for every $v\in V(G)$;
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\item for distinct $u,v\in V(G)$ and arbitrary $x\in F_u$, $y\in F_v$,
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\[
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\lambda(\dist_G(u,v)-1)+1
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\le
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\dist_H(x,y)
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\le
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\lambda(\dist_G(u,v)+2)-2;
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\]
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\item $\diam_H(F_v)\le2(\lambda-1)$ for every $v\in V(G)$.
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\end{enumerate}
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\end{definition}
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The particular constants in Definition~\ref{def:projection} are chosen because they are
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exact for the HMGHM tower. The proof below only needs a bounded additive
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slack after division by $\lambda$ and a strict gap between deployment and
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escape deadlines.
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For $U\subseteq V(G)$, let $B_G(U,r)$ be its closed radius-$r$
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neighborhood.
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\begin{theorem}[Abstract macro-ball occupation transfer]\label{thm:abstract-transfer}
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Let $G$ have $N$ vertices. Fix $d\ge2$, $R\ge2$, and constants $a,A_0>0$.
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Assume
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\begin{equation}\label{eq:scale}
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\sqrt N\le d^{R-2}<d\sqrt N,
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\qquad
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d^3\le\sqrt N,
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\end{equation}
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and
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\begin{align}
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|B_G(U,R-2)|
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&\ge a\min\{|U|d^{R-2},N\}
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&&\text{for every }U\subseteq V(G),\label{eq:lowergrowth}\\
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|B_G(v,R)|
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&\le A_0d^R
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&&\text{for every }v\in V(G).\label{eq:uppergrowth}
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\end{align}
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If $H$ admits a $(\lambda,P)$-occupation projection onto $G$, then
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\[
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\boxed{
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\cnum(H)
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\le
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C(a,A_0)P\bigl(d^3+\log(ePN)\bigr)\sqrt N.
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}
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\]
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The displayed number of cops captures the robber within at most $\lambda R$
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cop moves. In particular, the cop-number bound is independent of the scale
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factor $\lambda$; tower depth affects capture time but not the required bank
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size.
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\end{theorem}
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\begin{proof}
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Put
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\[
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\Theta=d^3+\log(ePN),
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\qquad
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\mu=\frac{AP\Theta}{\sqrt N},
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\]
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where $A$ is a sufficiently large constant depending only on $a,A_0$.
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Choose one canonical vertex $z^*\in F_z$ in every fiber. At $z^*$ place an
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independent Poisson number of cop tokens of mean $\mu$.
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For a possible robber fiber $F_v$, set
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\[
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X_v=\pi^{-1}(B_G(v,R)).
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\]
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By \eqref{eq:uppergrowth} and \eqref{eq:scale},
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\begin{equation}\label{eq:Qbound}
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Q_v:=|X_v|
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\le PA_0d^R
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<A_0Pd^3\sqrt N
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\le A_0P\Theta\sqrt N.
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\end{equation}
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We prove simultaneously for every $v$ that the sampled tokens can be matched
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distinctly to all vertices of $X_v$, with every assigned token based over a
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base vertex within distance $R-2$ of its target fiber.
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Let $S\subseteq X_v$, $|S|=s$, and put $U=\pi(S)$. Since every fiber has at
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most $P$ targets, $|U|\ge s/P$. Every token based over
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$Z=B_G(U,R-2)$ is adjacent in the assignment graph to at least one target in
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$S$.
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If $|U|d^{R-2}<N$, then the number of available tokens is Poisson with mean
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at least
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\[
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\mu a|U|d^{R-2}\ge Aa\Theta s.
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\]
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For a Poisson variable $Y$ of mean $\Lambda\ge Aa\Theta s$,
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\[
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\Pr(Y<s)\le e^{-\Lambda}\left(\frac{e\Lambda}{s}\right)^s.
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\]
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After increasing $A$, this is at most $(ePN)^{-4s}$. There are at most
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$N\binom{PN}{s}$ choices of a root and a target subset of size $s$, so summing
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over $s\ge1$ gives $o(1)$.
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If $|U|d^{R-2}\ge N$, then the available-token mean is at least
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\[
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\mu aN=AaP\Theta\sqrt N
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\ge\frac{Aa}{A_0}Q_v
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\]
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by \eqref{eq:Qbound}. Taking $A$ large and applying the same Poisson bound
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shows, after a union bound over at most $N2^{Q_v}$ target subsets, that no
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saturated Hall condition fails with probability $1-o(1)$. Here every
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saturated target set has
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\[
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s\ge |U|\ge\frac{N}{d^{R-2}}>\frac{\sqrt N}{d}\ge N^{1/3},
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\]
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so the polynomial prefactors are negligible.
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Hall's condition therefore holds simultaneously for all macro-balls with
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probability $1-o(1)$. The total number of sampled tokens is at most
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$2AP\Theta\sqrt N$ with probability $1-o(1)$, so a deterministic placement of
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the claimed size exists.
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It remains to compare deadlines. A target $y\in X_v$ lies over some
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$u\in B_G(v,R)$. Its assigned cop begins over $z$ with
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$\dist_G(z,u)\le R-2$. If $z\ne u$, the upper distortion bound gives travel
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time at most
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\[
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\lambda((R-2)+2)-2=\lambda R-2.
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\]
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If $z=u$, the fiber-diameter bound gives at most
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$2(\lambda-1)\le\lambda R-2$, since $R\ge2$.
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To leave $X_v$, the robber must enter a fiber over a base vertex at distance
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at least $R+1$ from $v$. The lower distortion bound makes this require at
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least
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\[
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\lambda((R+1)-1)+1=\lambda R+1
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\]
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steps. Every vertex of $X_v$ is occupied first, and the cop assigned to the
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robber's current vertex captures her.
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\end{proof}
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\begin{remark}[The quantifier needed from the random base]\label{rem:PW-uniform}
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The use of \eqref{eq:lowergrowth} is graph-uniform, not a per-source-set
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probability statement. In the dense theorem of Pra{\l}at and Wormald,
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condition~(i) of their deterministic Theorem~3.1 is explicitly quantified over
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\emph{every} source set and radius. Their Theorem~3.4 proves that a single
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$G(N,p)$ satisfies those hypotheses asymptotically almost surely; its proof
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unions over the bad source sets and concludes that the growth estimate holds
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simultaneously for all sets and radii \cite[Theorems~3.1 and~3.4]{PW}.
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Thus the external input has the quantifier order required by
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\cref{thm:abstract-transfer}.
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\end{remark}
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\section{The HMGHM replacement tower}
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For a vertex of degree $r$, the HMGHM replacement has one external port for
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every incident edge. The ports are partitioned into nearly equal classes,
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and for each pair of classes there is an internal vertex adjacent to every
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port in the two classes \cite[Section~2]{HMGHM}.
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\begin{lemma}[One-round port geometry]\label{lem:portgeometry}
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For every HMGHM replacement cloud of degree at least two:
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\begin{enumerate}[label=\textup{(\roman*)}]
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\item distinct ports are nonadjacent and have distance exactly two;
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\item every cloud vertex is within distance at most three of every specified
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port;
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\item the cloud diameter is at most four.
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\end{enumerate}
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\end{lemma}
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\begin{proof}
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Two ports in different classes share the internal vertex associated with
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their class pair. Two ports in the same class share any internal vertex
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associated with that class and another nonempty class. Since ports are
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mutually nonadjacent, their distance is exactly two.
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An internal vertex is adjacent to every port in either of two classes. If a
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specified port lies in neither class, travel to a port in one of the two
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classes, then through the internal vertex corresponding to that class and the
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specified port's class, and finally to the specified port. This takes three
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steps. The diameter bound follows by routing arbitrary endpoints through a
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specified port.
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\end{proof}
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Let
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\[
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G=G_0,G_1,\ldots,G_k=H
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\]
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be an iterated HMGHM tower, and let $\pi:V(H)\to V(G)$ map every final vertex
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to its original ancestor. Put
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\[
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F_v=\pi^{-1}(v),
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\qquad
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P=\max_v|F_v|,
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\qquad
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\lambda=3^k.
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\]
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\begin{theorem}[Exact normalized distortion]\label{thm:metric}
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The ancestry projection is a $(3^k,P)$-occupation projection. Explicitly,
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for distinct base vertices $u,v$, $r=\dist_G(u,v)$, and arbitrary
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$x\in F_u$, $y\in F_v$,
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\[
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\boxed{
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3^k(r-1)+1
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\le
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\dist_H(x,y)
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\le
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3^k(r+2)-2,
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}
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\]
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and
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\[
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\boxed{\diam_H(F_v)\le2(3^k-1).}
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\]
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\end{theorem}
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\begin{proof}
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For one round, a shortest path between distinct clouds uses $e\ge r$
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external edges. Between consecutive external edges it enters and leaves an
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intermediate cloud through distinct ports: otherwise it immediately traverses
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one external edge back. By Lemma~\ref{lem:portgeometry}, each intermediate port
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change costs at least two internal edges, and hence
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\[
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\dist_{G_1}(x,y)\ge e+2(e-1)\ge3r-2.
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\]
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For the upper bound, follow a base geodesic. Reaching the first prescribed
|
|
port costs at most three, each intermediate port change costs two, the
|
|
external edges cost $r$, and reaching the final endpoint costs at most three.
|
|
Thus
|
|
\[
|
|
\dist_{G_1}(x,y)\le3+r+2(r-1)+3=3r+4.
|
|
\]
|
|
The one-round fiber diameter is at most four.
|
|
|
|
The lower and upper affine recurrences are
|
|
\[
|
|
L_j(r)=3L_{j-1}(r)-2,
|
|
\qquad
|
|
U_j(r)=3U_{j-1}(r)+4,
|
|
\]
|
|
with $L_0(r)=U_0(r)=r$. Solving gives
|
|
\[
|
|
L_k(r)=3^k(r-1)+1,
|
|
\qquad
|
|
U_k(r)=3^k(r+2)-2.
|
|
\]
|
|
The diameter recurrence $D_j\le3D_{j-1}+4$, $D_0=0$, gives
|
|
$D_k\le2(3^k-1)$.
|
|
\end{proof}
|
|
|
|
The feature that matters is not the number of rounds but the normalized
|
|
additive error: after division by $3^k$, it remains two quotient layers. By
|
|
\cref{thm:abstract-transfer}, any other graph projection with the same three
|
|
properties inherits the same occupation-certificate transfer.
|
|
|
|
\section{Expansion retention under port-cloud replacement}
|
|
|
|
\begin{proposition}[Expansion under connected port replacement]\label{prop:port-exp}
|
|
Let $H$ be obtained from a base graph $G$ by replacing every vertex by a
|
|
connected cloud of order at most $L_0$, with distinct external ports for the
|
|
incident base edges. If $\iotae(G)$ is the edge-isoperimetric constant of
|
|
$G$, then
|
|
\[
|
|
\boxed{
|
|
\iotae(H)
|
|
\ge
|
|
\frac{1}{2L_0}
|
|
\min\left\{1,\frac{\iotae(G)}{L_0}\right\}.
|
|
}
|
|
\]
|
|
\end{proposition}
|
|
|
|
\begin{proof}
|
|
Let $S\subseteq V(H)$ with $0<|S|\le|H|/2$. In each cloud, classify the
|
|
majority side and let $\mathcal M$ be the total number of minority vertices. Since
|
|
every partially cut cloud is connected and has at most $L_0$ vertices, its
|
|
internal cut contributes at least one edge, so the total internal contribution
|
|
is at least $\mathcal M/L_0$.
|
|
|
|
Let $U$ be the set of base vertices whose clouds have majority in $S$, and
|
|
put $E=e_G(U,V(G)\setminus U)$. A base cut edge can fail to cross the lifted
|
|
cut only if one of its two ports is a minority vertex. Distinct base edges
|
|
use distinct ports, so at most $\mathcal M$ of the $E$ external cut edges fail. Hence
|
|
\[
|
|
e_H(S,V(H)\setminus S)
|
|
\ge
|
|
\frac{\mathcal M}{L_0}+\max\{0,E-\mathcal M\}
|
|
\ge
|
|
\frac{E+\mathcal M}{2L_0}.
|
|
\]
|
|
Apply the same majority accounting to $S$ or its complement, according as
|
|
$|U|\le|G|/2$ or not, to obtain
|
|
\[
|
|
\min\{|U|,|V(G)\setminus U|\}
|
|
\ge
|
|
\frac{|S|-\mathcal M}{L_0}.
|
|
\]
|
|
Thus
|
|
\[
|
|
E\ge\frac{\iotae(G)}{L_0}(|S|-\mathcal M),
|
|
\]
|
|
and consequently
|
|
\[
|
|
E+\mathcal M
|
|
\ge
|
|
\min\left\{1,\frac{\iotae(G)}{L_0}\right\}|S|.
|
|
\]
|
|
Combining the inequalities proves the proposition.
|
|
\end{proof}
|
|
|
|
The HMGHM cloud-size recurrence turns the one-round estimate into a
|
|
subpolynomial-loss statement in the polylogarithmic-degree regime. If $D$
|
|
is the initial maximum degree, $L_i$ is the largest cloud order in round $i$,
|
|
and $k$ rounds are used, HMGHM prove
|
|
\[
|
|
\prod_{i=0}^{k-1}L_i
|
|
\le C D^2(\log D)^{\log_2(11/5)},
|
|
\qquad
|
|
2^k=O(\log D).
|
|
\]
|
|
Iterating Proposition~\ref{prop:port-exp} therefore gives the following.
|
|
|
|
\begin{corollary}[Expansion retained by HMGHM reduction]\label{cor:exp-retention}
|
|
Let $H$ be the final subcubic graph obtained from a connected graph $G$ of
|
|
maximum degree $D\ge4$. Then
|
|
\[
|
|
\boxed{
|
|
h(H)
|
|
\ge
|
|
\frac{\iotae(G)}{C D^4(\log D)^{\kappa}},
|
|
\qquad
|
|
\kappa=1+2\log_2(11/5)<3.28.
|
|
}
|
|
\]
|
|
In particular, if $D=|G|^{o(1)}$ and $\iotae(G)=|G|^{-o(1)}$, then
|
|
$|H|=|G|^{1+o(1)}$ and $h(H)=|H|^{-o(1)}$.
|
|
\end{corollary}
|
|
|
|
\begin{proof}
|
|
At every round $\iotae(G_i)\le D_i\le L_i$, so
|
|
Proposition~\ref{prop:port-exp} gives
|
|
\[
|
|
\iotae(G_{i+1})\ge\frac{\iotae(G_i)}{2L_i^2}.
|
|
\]
|
|
Thus
|
|
\[
|
|
\iotae(H)
|
|
\ge
|
|
\frac{\iotae(G)}{2^k(\prod_iL_i)^2}
|
|
\ge
|
|
\frac{\iotae(G)}{C D^4(\log D)^\kappa}.
|
|
\]
|
|
Since $H$ has maximum degree at most three, its vertex expansion is at least
|
|
one third of its edge expansion. The order statement follows from the same
|
|
cloud-product bound.
|
|
\end{proof}
|
|
|
|
\begin{remark}[Relation to replacement products]
|
|
The regular replacement-product literature proves stronger spectral
|
|
conclusions under much stronger hypotheses on the clouds; see, for example,
|
|
Reingold--Vadhan--Wigderson \cite{RVW}. Proposition~\ref{prop:port-exp} allows arbitrary
|
|
connected, nonuniform clouds and consequently gives only a crude
|
|
isoperimetric estimate. No novelty claim is made here beyond this precise
|
|
form without a fuller graph-substitution review.
|
|
\end{remark}
|
|
|
|
\section{A polylogarithmically tight square-root family}
|
|
|
|
Take
|
|
\[
|
|
d=(\log N)^4,
|
|
\qquad
|
|
p=\frac{d}{N-1},
|
|
\qquad
|
|
G\sim G(N,p).
|
|
\]
|
|
Iterate the HMGHM replacement until the graph $H$ is subcubic, and write
|
|
$M=|H|$.
|
|
|
|
With high probability, $\Delta(G)\le2d$. By
|
|
Remark~\ref{rem:PW-uniform}, the dense Pra{\l}at--Wormald theorem supplies the
|
|
uniform lower growth in \eqref{eq:lowergrowth}; in the volume range used here
|
|
it also supplies the upper growth in \eqref{eq:uppergrowth} \cite{PW}. Choose
|
|
$R$ minimally so that $d^{R-2}\ge\sqrt N$. Since $d$ is polylogarithmic,
|
|
\eqref{eq:scale} holds.
|
|
|
|
HMGHM give, both globally and along one ancestry fiber,
|
|
\begin{equation}\label{eq:Pbound}
|
|
P\le C d^2(\log d)^{1.14},
|
|
\qquad
|
|
N\le M\le PN.
|
|
\end{equation}
|
|
Their shadow strategy gives $\cnum(H)\ge\cnum(G)$, and the random-graph lower
|
|
bound of Bollob\'as--Kun--Leader used in their argument yields
|
|
\[
|
|
\cnum(G)
|
|
\ge
|
|
d^{-2}N^{\frac12-\frac{9}{2\log\log d}}.
|
|
\]
|
|
The abstract transfer theorem gives
|
|
\[
|
|
\cnum(H)
|
|
\le
|
|
CP\bigl(d^3+\log(ePN)\bigr)\sqrt N
|
|
\le
|
|
\sqrt M\,(\log M)^{20+o(1)}.
|
|
\]
|
|
Using $d=(\log N)^4$ and $M=N(\log N)^{O(1)}$ in the lower bound gives the
|
|
following more informative formulation.
|
|
|
|
\begin{theorem}[Quantitative HMGHM hard family]\label{thm:hardfamily}
|
|
There is a sequence of connected subcubic graphs $H$, of order $M\to\infty$,
|
|
for which
|
|
\[
|
|
\boxed{
|
|
M^{\frac12-\frac{9+o(1)}{2\log\log\log M}}
|
|
\le
|
|
\cnum(H)
|
|
\le
|
|
\sqrt M\,(\log M)^{20+o(1)}.
|
|
}
|
|
\]
|
|
The upper bound is $\sqrt M$ times a polylogarithmic factor. The lower
|
|
exponent tends to $1/2$ only at a triple-logarithmic rate.
|
|
\end{theorem}
|
|
|
|
The base edge expansion is $\Omega(d)$ with high probability. Since
|
|
$d=\operatorname{polylog}N$, Corollary~\ref{cor:exp-retention} gives
|
|
\[
|
|
h(H)\ge (\log M)^{-O(1)}=M^{-o(1)}.
|
|
\]
|
|
This is exactly the degree regime in which the retention factor is
|
|
informative; for polynomial initial degree the crude $D^4$ loss can be
|
|
vacuous.
|
|
|
|
\begin{corollary}[Square-root weak-expander family]\label{cor:weakexpander}
|
|
There are connected subcubic graphs satisfying
|
|
\[
|
|
h(H)\ge M^{-o(1)}
|
|
\qquad\text{and}\qquad
|
|
\cnum(H)=M^{1/2+o(1)}.
|
|
\]
|
|
More precisely, they obey the two-sided bounds of
|
|
\cref{thm:hardfamily}.
|
|
\end{corollary}
|
|
|
|
\begin{remark}[What is forced, and what is achieved]\label{rem:robustness-endpoint}
|
|
By \cref{prop:adaptive-core}, polynomially weak expansion is a natural
|
|
robustness window for weak Meyniel. The HMGHM lower bound alone already
|
|
forces every proposed estimate
|
|
\[
|
|
\cnum(J)\le C\phi^{-p}|J|^{1-\eps+o(1)}
|
|
\qquad(h(J)\ge\phi=|J|^{-o(1)})
|
|
\]
|
|
to have $\eps\le1/2$. That restriction predates the upper transfer proved
|
|
here. The new conclusion is that the known stressing family itself achieves
|
|
the endpoint order $\sqrt M$ up to polylogarithmic factors, so this family is
|
|
not an obstruction to a Meyniel-strength theorem on polynomially weak
|
|
subcubic expanders.
|
|
\end{remark}
|
|
|
|
\section{Why chase strategies need not transfer}
|
|
|
|
The abstract theorem deliberately transfers a strategy class, not arbitrary
|
|
cop number. The smallest example explains the distinction. For a degree-two
|
|
vertex, one HMGHM cloud is a three-vertex path. Replacing every vertex of
|
|
$C_3$ therefore produces $C_9$. But
|
|
\[
|
|
\cnum(C_3)=1,
|
|
\qquad
|
|
\cnum(C_9)=2.
|
|
\]
|
|
The one-cop win on $C_3$ is a direct chase/dismantling phenomenon. The
|
|
subdivision-like stretching destroys it. By contrast, an occupation
|
|
certificate is synchronized to a deadline: the replacement tower stretches
|
|
the cops' travel and the robber's escape by the same factor, and the bounded
|
|
normalized additive slack preserves a strict margin. The examples
|
|
$K_4$ and the diamond graph exhibit the same one-round increase, so the issue
|
|
is structural rather than peculiar to one cycle.
|
|
|
|
\section{A one-shot occupation barrier}
|
|
|
|
\begin{definition}[Universal one-shot occupation number]\label{def:occ}
|
|
For a connected graph $G$ and integer $R\ge0$, let $\Occ_R(G)$ be the minimum
|
|
size of a finite set $X$ of distinct cop tokens, equipped with a position map
|
|
$p:X\to V(G)$, such that for every $v\in V(G)$ there is an injection
|
|
\[
|
|
f_v:B_G(v,R)\longrightarrow X
|
|
\]
|
|
with
|
|
\[
|
|
\dist_G(u,p(f_v(u)))\le R
|
|
\qquad\text{for every }u\in B_G(v,R).
|
|
\]
|
|
Different tokens may have the same initial position. After learning the
|
|
robber's starting vertex, the common prepositioned bank can occupy her entire
|
|
radius-$R$ ball within $R$ moves.
|
|
\end{definition}
|
|
|
|
\begin{remark}[Why the target and deadline are natural]
|
|
If the robber starts at $v$, she needs at least $R+1$ robber moves to leave
|
|
$B_G(v,R)$. Occupying that whole ball within $R$ cop moves is therefore the
|
|
canonical one-shot certificate: every vertex she could still occupy is filled
|
|
before her first possible escape. The parameter $\Occ_R$ measures this
|
|
specific strategy class, not ordinary cop number.
|
|
\end{remark}
|
|
|
|
\begin{theorem}[Counting barrier]\label{thm:occ-lower}
|
|
Every connected graph satisfies
|
|
\[
|
|
\boxed{
|
|
\Occ_R(G)
|
|
\ge
|
|
\frac{\sum_{v\in V(G)}|B_G(v,R)|}
|
|
{\max_{x\in V(G)}|B_G(x,2R)|}.
|
|
}
|
|
\]
|
|
In particular, if $G$ is vertex-transitive, then
|
|
\[
|
|
\boxed{
|
|
\Occ_R(G)
|
|
\ge
|
|
|V(G)|\frac{|B_G(o,R)|}{|B_G(o,2R)|}.
|
|
}
|
|
\]
|
|
\end{theorem}
|
|
|
|
\begin{proof}
|
|
Fix a feasible multiset $X$. For each possible robber start $v$, every cop
|
|
token used by the injection $f_v$ lies in $B_G(v,2R)$, by the triangle
|
|
inequality. Hence at least $|B_G(v,R)|$ tokens of $X$ lie in $B_G(v,2R)$.
|
|
Summing over $v$, the number of incident pairs $(v,x)$ with
|
|
$x\in X\cap B_G(v,2R)$ is at least
|
|
$\sum_v|B_G(v,R)|$.
|
|
|
|
A fixed token based at $x$ is counted only for starts
|
|
$v\in B_G(x,2R)$, at most $\max_y|B_G(y,2R)|$ times. Therefore
|
|
\[
|
|
|X|\max_y|B_G(y,2R)|
|
|
\ge
|
|
\sum_v|B_G(v,R)|,
|
|
\]
|
|
which proves the claim.
|
|
\end{proof}
|
|
|
|
The theorem identifies the exact growth ratio demanded by one-shot
|
|
occupation. A polynomial saving from the trivial $|V(G)|$ bound requires
|
|
polynomial amplification from radius $R$ to radius $2R$.
|
|
|
|
We now give subcubic witnesses showing that polynomially weak expansion does
|
|
not imply such amplification.
|
|
|
|
\begin{definition}[The cubic truncated torus]\label{def:Qt}
|
|
For $L\ge5$, let $Q_L$ have vertex set
|
|
\[
|
|
(\mathbb Z/L\mathbb Z)^2\times\mathbb Z/4\mathbb Z.
|
|
\]
|
|
Inside each fiber $(x,y)\times\mathbb Z/4\mathbb Z$, join the four vertices in
|
|
a cycle. Add the external edges
|
|
\[
|
|
(x,y,0)(x,y+1,2)
|
|
\qquad\text{and}\qquad
|
|
(x,y,1)(x+1,y,3)
|
|
\]
|
|
for every $(x,y)$. Equivalently, $(x,y,2)$ receives its external edge from
|
|
$(x,y-1,0)$, and $(x,y,3)$ receives its external edge from $(x-1,y,1)$.
|
|
Thus every vertex has two internal cycle neighbors and one external neighbor,
|
|
and $Q_L$ is the four-cycle port replacement of the square torus
|
|
$C_L\square C_L$.
|
|
\end{definition}
|
|
|
|
Every vertex of $Q_L$ has degree three, and the construction embeds on the
|
|
torus by replacing each base vertex inside a small disk. Its order is
|
|
$4L^2$.
|
|
|
|
|
|
|
|
\begin{lemma}[Vertex transitivity and explicit doubling of $Q_L$]\label{lem:doubling}
|
|
The graph $Q_L$ is vertex-transitive and, for every vertex $x$ and radius
|
|
$R\ge0$,
|
|
\[
|
|
|B_{Q_L}(x,2R)|\le 5500\,|B_{Q_L}(x,R)|.
|
|
\]
|
|
\end{lemma}
|
|
|
|
\begin{proof}
|
|
Translations in the first two coordinates are automorphisms. The map
|
|
\[
|
|
\rho(x,y,i)=(y,-x,i+1)
|
|
\]
|
|
(with coordinates interpreted cyclically) preserves internal cycle edges and
|
|
interchanges the two external edge directions. Translations together with
|
|
$\rho$ act transitively.
|
|
|
|
Let $T_L=C_L\square C_L$ and project $(x,y,i)$ to $(x,y)$. Projection does
|
|
not increase distance, so
|
|
\[
|
|
|B_{Q_L}(x,2R)|\le4|B_{T_L}(\pi x,2R)|
|
|
\le4\min\{L,4R+1\}^2.
|
|
\]
|
|
Conversely, from an arbitrary cloud vertex one can enter the required port in
|
|
at most two internal moves and then lift each base step using at most three
|
|
moves. Hence, with $r=\lfloor(R-2)/3\rfloor$ for $R\ge2$,
|
|
\[
|
|
|B_{Q_L}(x,R)|\ge |B_{T_L}(\pi x,r)|.
|
|
\]
|
|
The coordinate box of cyclic radius $\lfloor r/2\rfloor$ lies inside the
|
|
$\ell_1$ ball, so
|
|
\[
|
|
|B_{T_L}(\pi x,r)|
|
|
\ge \min\{L,2\lfloor r/2\rfloor+1\}^2.
|
|
\]
|
|
For $R<10$, the upper bound is at most $4\cdot37^2<5500$ and the denominator
|
|
is at least one. For $R\ge10$, one has $r\ge R/6$ and
|
|
$2\lfloor r/2\rfloor+1\ge r$, whence the ratio is at most
|
|
$4\cdot30^2<5500$. This proves the displayed constant.
|
|
\end{proof}
|
|
|
|
\begin{theorem}[Cubic one-shot barrier]\label{thm:cubic-barrier}
|
|
For $L\ge5$, the connected cubic graphs $Q_L$, with $M=|Q_L|=4L^2$, satisfy
|
|
\[
|
|
\boxed{
|
|
h(Q_L)=\Theta(M^{-1/2}),
|
|
\qquad
|
|
\cnum(Q_L)\le3,
|
|
\qquad
|
|
\Occ_R(Q_L)\ge \frac{M}{5500}
|
|
\quad\text{for every }R\ge0.
|
|
}
|
|
\]
|
|
\end{theorem}
|
|
|
|
\begin{proof}
|
|
The square torus has edge and vertex expansion $\Theta(1/L)$ by the discrete
|
|
torus isoperimetric inequality \cite{BL}. Applying
|
|
Proposition~\ref{prop:port-exp} with cloud size four gives the matching lower
|
|
bound for $Q_L$; lifting a coordinate slab of width $\lfloor L/2\rfloor$
|
|
gives the upper bound for every $L$. Since $Q_L$ is cubic, edge and vertex
|
|
expansion differ by at most a constant factor.
|
|
Thus $h(Q_L)=\Theta(1/L)=\Theta(M^{-1/2})$.
|
|
|
|
The graph $Q_L$ is toroidal, and every toroidal graph has cop number at most
|
|
three \cite{Lehner}. Vertex transitivity, Theorem~\ref{thm:occ-lower}, and
|
|
Lemma~\ref{lem:doubling} give
|
|
\[
|
|
\Occ_R(Q_L)
|
|
\ge
|
|
M\frac{|B_{Q_L}(x,R)|}{|B_{Q_L}(x,2R)|}
|
|
\ge \frac{M}{5500}.
|
|
\]
|
|
The finite audit suggests that the optimal asymptotic constant is $1/4$, but
|
|
that sharpening is not needed here.
|
|
\end{proof}
|
|
|
|
\subsection{The barrier throughout every polynomial expansion window}
|
|
|
|
For integers $k\ge2$ and $L\ge4$, write
|
|
\[
|
|
T_{L,k}=\underbrace{C_L\square\cdots\square C_L}_{k\text{ factors}}.
|
|
\]
|
|
Its order is $m=L^k$, its degree is $2k$, and its metric is the cyclic
|
|
$\ell_1$ metric.
|
|
|
|
\begin{lemma}[Uniform doubling of Cartesian tori]\label{lem:ktorus-doubling}
|
|
For every fixed $k\ge2$, every $L\ge4$, every vertex $x$, and every radius
|
|
$R\ge0$,
|
|
\[
|
|
|B_{T_{L,k}}(x,2R)|\le (5k)^k|B_{T_{L,k}}(x,R)|.
|
|
\]
|
|
\end{lemma}
|
|
|
|
\begin{proof}
|
|
Every coordinate of a point in $B(x,2R)$ has cyclic distance at most $2R$, so
|
|
\[
|
|
|B(x,2R)|\le \min\{L,4R+1\}^k.
|
|
\]
|
|
The coordinate box in which every coordinate has cyclic distance at most
|
|
$\lfloor R/k\rfloor$ lies in $B(x,R)$, and therefore
|
|
\[
|
|
|B(x,R)|\ge\min\{L,2\lfloor R/k\rfloor+1\}^k.
|
|
\]
|
|
If $R<k$, the ratio is at most $(4k+1)^k$. If $R\ge k$, then
|
|
$2\lfloor R/k\rfloor+1\ge R/k$ and $4R+1\le5R$; taking the minima with $L$
|
|
does not increase their ratio beyond $5k$. The claim follows.
|
|
\end{proof}
|
|
|
|
\begin{theorem}[Full-window toroidal barrier]\label{thm:full-window}
|
|
For every fixed $k\ge2$, the graphs $T_{L,k}$ satisfy
|
|
\[
|
|
\boxed{
|
|
h(T_{L,k})=\Theta_k(m^{-1/k}),
|
|
\qquad
|
|
\cnum(T_{L,k})=k+1,
|
|
\qquad
|
|
\Occ_R(T_{L,k})\ge (5k)^{-k}m
|
|
\quad\text{for every }R\ge0.
|
|
}
|
|
\]
|
|
Consequently, for every $\delta>0$ there is a constant-degree graph family
|
|
with
|
|
\[
|
|
h(G)\ge |G|^{-\delta},
|
|
\qquad
|
|
c(G)=O_\delta(1),
|
|
\qquad
|
|
\Occ_R(G)=\Omega_\delta(|G|)
|
|
\quad\text{for every radius }R.
|
|
\]
|
|
\end{theorem}
|
|
|
|
\begin{proof}
|
|
The discrete-torus edge-isoperimetric inequality gives order $1/L$
|
|
\cite{BL}. Since $T_{L,k}$ has degree $2k$, edge boundary and external
|
|
vertex boundary differ by at most the fixed factor $2k$; a coordinate slab of
|
|
width $\lfloor L/2\rfloor$ supplies the matching upper bound. Hence
|
|
$h(T_{L,k})=\Theta_k(1/L)=\Theta_k(m^{-1/k})$. Neufeld and Nowakowski proved
|
|
that a Cartesian product of $k$ cycles, each of length at least four, has cop
|
|
number exactly $k+1$ \cite{NeufeldNowakowski}. Since the torus is
|
|
vertex-transitive, Theorem~\ref{thm:occ-lower} and
|
|
Lemma~\ref{lem:ktorus-doubling} give the occupation lower bound.
|
|
|
|
Given $\delta>0$, choose $k=\max\{2,\lfloor1/\delta\rfloor+1\}$. Then $1/k<\delta$, so for sufficiently large $m$ the expansion
|
|
lower bound $h(T_{L,k})\ge m^{-\delta}$ holds after absorbing the fixed
|
|
$k$-dependent constant.
|
|
\end{proof}
|
|
|
|
\begin{remark}[The sharp metric constant]
|
|
For fixed $k$, choose radii $1\ll R\ll L$. Lattice-point asymptotics for the
|
|
$\ell_1$ ball give
|
|
\[
|
|
\frac{|B_{T_{L,k}}(x,2R)|}{|B_{T_{L,k}}(x,R)|}=2^k+o(1).
|
|
\]
|
|
Thus no uniform doubling constant below $2^k$ is possible, and the counting
|
|
bound of \cref{thm:occ-lower} approaches the natural fraction $2^{-k}m$ on
|
|
these local radii. The explicit constant $(5k)^{-k}$ is chosen only for a
|
|
short all-radii proof.
|
|
\end{remark}
|
|
|
|
\begin{corollary}[No expansion-only one-shot theorem]\label{cor:no-exp-occ}
|
|
For every $\delta>0$, there is no implication of the form
|
|
\[
|
|
h(G)\ge |G|^{-\delta}
|
|
\quad\Longrightarrow\quad
|
|
\Occ_R(G)\le |G|^{1-\eps}
|
|
\text{ for some radius $R$}
|
|
\]
|
|
with any fixed $\eps>0$, even when the maximum degree is bounded by a constant
|
|
depending only on $\delta$.
|
|
\end{corollary}
|
|
|
|
\begin{proof}
|
|
Take the family from \cref{thm:full-window}. It lies in the prescribed
|
|
expansion window, while $\Occ_R(G)=\Omega_\delta(|G|)$ for every $R$.
|
|
\end{proof}
|
|
|
|
\begin{remark}[Architectural meaning]
|
|
The logical obstruction is not that tori are difficult pursuit instances;
|
|
they are not. Rather, \cref{thm:occ-lower} makes every vertex-transitive
|
|
bounded-doubling graph expensive for one-shot occupation, and bounded
|
|
doubling is compatible with every polynomial weak-expansion window by
|
|
\cref{thm:full-window}. The tori
|
|
certify that the hypothesis class contains such graphs while coordinate-wise
|
|
shadowing still uses exactly $k+1$ cops. Therefore ball amplification is the
|
|
wrong invariant for adaptive pursuit, not merely a poor description of one
|
|
particular easy family. The cubic family $Q_L$ records that the same
|
|
separation already occurs in maximum degree three at exponent $1/2$.
|
|
\end{remark}
|
|
|
|
\section{Outlook}
|
|
|
|
For vertex expansion $h(G)\ge\phi$, iterating the elementary growth factor
|
|
$1+\phi$ reaches global scale after $O(\phi^{-1}\log |G|)$ layers. The same
|
|
iteration gives the standard diameter bound of that order; these are two
|
|
forms of the same calculation, not independent evidence. The strategic
|
|
consequence is that, when $\phi=|G|^{-a}$, an amplification-based pursuit
|
|
scheme must operate over a full-traversal timescale $\Theta(|G|^a\log|G|)$.
|
|
|
|
The present paper separates three phenomena:
|
|
\begin{enumerate}[label=\textup{(\arabic*)}]
|
|
\item bounded normalized metric distortion preserves a strong occupation
|
|
certificate through degree reduction;
|
|
\item the HMGHM stressing family itself meets the square-root endpoint up to
|
|
polylogarithmic factors;
|
|
\item one-shot occupation is nevertheless incapable of proving a universal
|
|
robustness theorem throughout any polynomial weak-expansion window, even on
|
|
bounded-degree graphs with constant cop number; a cubic instance already
|
|
appears at exponent $1/2$.
|
|
\end{enumerate}
|
|
The remaining universal question is therefore an adaptive one. On the tori,
|
|
ball growth carries essentially no information about pursuit cost; product
|
|
structure instead supports coordinate-wise shadowing. What geometric or
|
|
combinatorial quantity replaces product coordinates on a general
|
|
polynomially weak expander? Equivalently, can a capacitated, correlated, or
|
|
deferred witness system reuse the same cop resources over polynomially many
|
|
weak-growth layers, or must every such one-traversal certificate incur
|
|
polynomial congestion?
|
|
|
|
\section*{Acknowledgments}
|
|
The author is grateful to Anthony Clow, Peter Bradshaw, Bojan Mohar, and
|
|
Florian Lehner for work and perspectives that helped shape the questions
|
|
addressed here. Additional acknowledgments will be added in a later version.
|
|
The author welcomes corrections concerning priority, related
|
|
graph-substitution inequalities, and the scope of the occupation framework.
|
|
|
|
\section*{Audit and reproducibility}
|
|
|
|
The metric inequalities were independently tested on HMGHM towers rebuilt
|
|
from the published gadget description, including structured base graphs not
|
|
used in the original audit. The Hall inequalities and timing margins were
|
|
checked numerically, and exact small replacement games were solved by
|
|
retrograde analysis. The toroidal barrier audit computes exact ball profiles of
|
|
$C_L^{\square k}$ by convolving cyclic distance distributions, verifies the
|
|
$(5k)^k$ doubling bound for $k=2,3,4,5$, constructs $Q_L$, checks cubicity,
|
|
connectivity, and the displayed rotation automorphism, and evaluates the
|
|
counting lower bound at every radius. No theorem depends on the
|
|
computations.
|
|
|
|
\begin{thebibliography}{99}
|
|
|
|
\bibitem{AignerFromme}
|
|
M.~Aigner and M.~Fromme,
|
|
\emph{A game of cops and robbers},
|
|
Discrete Applied Mathematics 8 (1984), 1--12.
|
|
|
|
\bibitem{CurrentFrontier}
|
|
P.~Bose, L.~Esperet, J.~Hodor, G.~Joret, P.~Micek, and C.~Rambaud,
|
|
\emph{Cops and robber in graphs with bounded vertex cover number},
|
|
arXiv:2602.07435, 2026.
|
|
|
|
\bibitem{BHMS}
|
|
P.~Bradshaw, S.~A. Hosseini, B.~Mohar, and L.~Stacho,
|
|
\emph{On the cop number of graphs of high girth},
|
|
Journal of Graph Theory 102 (2023), 15--34; arXiv:2005.10849.
|
|
|
|
\bibitem{BKL}
|
|
B.~Bollob\'as, G.~Kun, and I.~Leader,
|
|
\emph{Cops and robbers in a random graph},
|
|
Journal of Combinatorial Theory, Series B 103 (2013), 226--236.
|
|
|
|
\bibitem{BL}
|
|
B.~Bollob\'as and I.~Leader,
|
|
\emph{An isoperimetric inequality on the discrete torus},
|
|
SIAM Journal on Discrete Mathematics 3 (1990), 32--37.
|
|
|
|
\bibitem{Clow}
|
|
A.~Clow,
|
|
\emph{Expanders satisfy the weak Meyniel conjecture},
|
|
withdrawn preprint, arXiv:2311.13792, 2023.
|
|
|
|
\bibitem{HMGHM}
|
|
S.~A. Hosseini, B.~Mohar, and S.~Gonzalez Hermosillo de la Maza,
|
|
\emph{Meyniel's conjecture on graphs of bounded degree},
|
|
Journal of Graph Theory 97 (2021), 401--407; arXiv:1912.06957.
|
|
|
|
\bibitem{Lehner}
|
|
F.~Lehner,
|
|
\emph{On the cop number of toroidal graphs},
|
|
Journal of Combinatorial Theory, Series B 151 (2021), 250--262.
|
|
|
|
\bibitem{LuPeng}
|
|
L.~Lu and X.~Peng,
|
|
\emph{On Meyniel's conjecture of the cop number},
|
|
Journal of Graph Theory 71 (2012), 192--205.
|
|
|
|
\bibitem{NeufeldNowakowski}
|
|
S.~Neufeld and R.~Nowakowski,
|
|
\emph{A game of cops and robbers played on products of graphs},
|
|
Discrete Mathematics 186 (1998), 253--268.
|
|
|
|
\bibitem{PW}
|
|
P.~Pra{\l}at and N.~Wormald,
|
|
\emph{Meyniel's conjecture holds for random graphs},
|
|
Random Structures \& Algorithms 48 (2016), 396--421; arXiv:1301.2841.
|
|
|
|
\bibitem{RVW}
|
|
O.~Reingold, S.~Vadhan, and A.~Wigderson,
|
|
\emph{Entropy waves, the zig-zag graph product, and new constant-degree
|
|
expanders},
|
|
Annals of Mathematics 155 (2002), 157--187.
|
|
|
|
\bibitem{ScottSudakov}
|
|
A.~Scott and B.~Sudakov,
|
|
\emph{A bound for the cops and robbers problem},
|
|
SIAM Journal on Discrete Mathematics 25 (2011), 1438--1442.
|
|
|
|
\end{thebibliography}
|
|
|
|
\end{document}
|