Independent set and matching permutations

Let $G$ be a graph $G$ whose largest independent set has size $m$. A permutation $\pi$ of $\{1, \ldots, m\}$ is an {\em independent set permutation} of $G$ if $$ a_{\pi(1)}(G) \leq a_{\pi(2)}(G) \leq \cdots \leq a_{\pi(m)}(G) $$ where $a_k(G)$ is the number of independent sets of size...

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Bibliographic Details
Published in	arXiv.org
Main Authors	Ball, Taylor, Galvin, David, Hyry, Catherine, Weingartner, Kyle
Format	Paper
Language	English
Published	Ithaca Cornell University Library, arXiv.org 13.07.2021
Subjects	Apexes Graph matching Graph theory Permutations State court decisions Upper bounds US exports
Online Access	Get full text

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Summary:	Let $G$ be a graph $G$ whose largest independent set has size $m$. A permutation $\pi$ of $\{1, \ldots, m\}$ is an {\em independent set permutation} of $G$ if $$ a_{\pi(1)}(G) \leq a_{\pi(2)}(G) \leq \cdots \leq a_{\pi(m)}(G) $$ where $a_k(G)$ is the number of independent sets of size $k$ in $G$. In 1987 Alavi, Malde, Schwenk and Erdős proved that every permutation of $\{1, \ldots, m\}$ is an independent set permutation of some graph with $\alpha(G)=m$, i.e. with largest independent set having size $m$. They raised the question of determining, for each $m$, the smallest number $f(m)$ such that every permutation of $\{1, \ldots, m\}$ is an independent set permutation of some graph with $\alpha(G)=m$ and with at most $f(m)$ vertices, and they gave an upper bound on $f(m)$ of roughly $m^{2m}$. Here we settle the question, determining $f(m)=m^m$, and make progress on a related question, that of determining the smallest order such that every permutation of $\{1, \ldots, m\}$ is the {\em unique} independent set permutation of some graph of at most that order. More generally we consider an extension of independent set permutations to weak orders, and extend Alavi et al.'s main result to show that every weak order on $\{1, \ldots, m\}$ can be realized by the independent set sequence of some graph with $\alpha(G)=m$ and with at most $m^{m+2}$ vertices. Alavi et al. also considered {\em matching permutations}, defined analogously to independent set permutations. They observed that not every permutation of $\{1,\ldots,m\}$ is a matching permutation of some graph with largest matching having size $m$, putting an upper bound of $2^{m-1}$ on the number of matching permutations of $\{1,\ldots,m\}$. Confirming their speculation that this upper bound is not tight, we improve it to $O(2^m/\sqrt{m})$.
ISSN:	2331-8422

Summary:	Let \(G\) be a graph \(G\) whose largest independent set has size \(m\). A permutation \(\pi\) of \(\{1, \ldots, m\}\) is an {\em independent set permutation} of \(G\) if $$ a_{\pi(1)}(G) \leq a_{\pi(2)}(G) \leq \cdots \leq a_{\pi(m)}(G) $$ where \(a_k(G)\) is the number of independent sets of size \(k\) in \(G\). In 1987 Alavi, Malde, Schwenk and Erdős proved that every permutation of \(\{1, \ldots, m\}\) is an independent set permutation of some graph with \(\alpha(G)=m\), i.e. with largest independent set having size \(m\). They raised the question of determining, for each \(m\), the smallest number \(f(m)\) such that every permutation of \(\{1, \ldots, m\}\) is an independent set permutation of some graph with \(\alpha(G)=m\) and with at most \(f(m)\) vertices, and they gave an upper bound on \(f(m)\) of roughly \(m^{2m}\). Here we settle the question, determining \(f(m)=m^m\), and make progress on a related question, that of determining the smallest order such that every permutation of \(\{1, \ldots, m\}\) is the {\em unique} independent set permutation of some graph of at most that order. More generally we consider an extension of independent set permutations to weak orders, and extend Alavi et al.'s main result to show that every weak order on \(\{1, \ldots, m\}\) can be realized by the independent set sequence of some graph with \(\alpha(G)=m\) and with at most \(m^{m+2}\) vertices. Alavi et al. also considered {\em matching permutations}, defined analogously to independent set permutations. They observed that not every permutation of \(\{1,\ldots,m\}\) is a matching permutation of some graph with largest matching having size \(m\), putting an upper bound of \(2^{m-1}\) on the number of matching permutations of \(\{1,\ldots,m\}\). Confirming their speculation that this upper bound is not tight, we improve it to \(O(2^m/\sqrt{m})\).
ISSN:	2331-8422