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Linear recoloring diameter of degenerate chordal graphs and bounded treewidth graphs

Yichen Wang, Mei Lu

Abstract

Let $G$ be a graph on $n$ vertices and $t$ an integer. The reconfiguration graph of $G$, denoted by $R_t(G)$, consists of all $t$-colorings of $G$ and two $t$-colorings are adjacent if they differ on exactly one vertex. The $t$-recoloring diameter of $G$ is the diameter of $R_t(G)$. For a $d$-degenerate graph $G$, $R_t(G)$ is connected when $t \ge d+2$~(Dyer et al., 2006). Furthermore, the $t$-recoloring diameter is $O(n^2)$ when $t \ge 3(d+1)/2$~(Bousquet et al., 2022), and it is $O(n)$ when $t \ge 2d+2$~(Bousquet and Perarnau, 2016). For a $d$-degenerate and chordal graph $G$, the $t$-recoloring diameter of $G$ is $O(n^2)$ when $t \ge d+2$~(Bonamy et al. 2014). If $G$ is a graph of treewidth at most $k$, then $G$ is also $k$-degenerate, and the previous results hold. Moreover, when $t \ge k+2$, the $t$-recoloring diameter is $O(n^2)$~(Bonamy and Bousquet, 2013). When $k=2$, the $t$-recoloring diameter of $G$ is linear when $t \ge 5$~(Bartier, Bousquet and Heinrich, 2021) and the result is tight. In this paper, we prove that if $G$ is $d$-degenerate and chordal, then the $t$-recoloring diameter of $G$ is $O(n)$ when $t \ge 2d+1$. Moreover, if the treewidth of $G$ is at most $k$, then the $t$-recoloring diameter is $O(n)$ when $t \ge 2k+1$. This result is a generalization of the previous results on graphs of treewidth at most two.

Linear recoloring diameter of degenerate chordal graphs and bounded treewidth graphs

Abstract

Let be a graph on vertices and an integer. The reconfiguration graph of , denoted by , consists of all -colorings of and two -colorings are adjacent if they differ on exactly one vertex. The -recoloring diameter of is the diameter of . For a -degenerate graph , is connected when ~(Dyer et al., 2006). Furthermore, the -recoloring diameter is when ~(Bousquet et al., 2022), and it is when ~(Bousquet and Perarnau, 2016). For a -degenerate and chordal graph , the -recoloring diameter of is when ~(Bonamy et al. 2014). If is a graph of treewidth at most , then is also -degenerate, and the previous results hold. Moreover, when , the -recoloring diameter is ~(Bonamy and Bousquet, 2013). When , the -recoloring diameter of is linear when ~(Bartier, Bousquet and Heinrich, 2021) and the result is tight. In this paper, we prove that if is -degenerate and chordal, then the -recoloring diameter of is when . Moreover, if the treewidth of is at most , then the -recoloring diameter is when . This result is a generalization of the previous results on graphs of treewidth at most two.

Paper Structure

This paper contains 5 sections, 6 theorems, 9 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Best choice algorithm
  4. Proof of Theorem \ref{['thm: 2k+1 linear']} and Corollary \ref{['cor: 2k+1 linear']}
  5. Discussion

Key Result

Theorem 1.2

Let $G$ be a $d$-degenerate and chordal graph and $t \ge 2d+1$. Given any two $t$-colorings $\alpha, \beta$ of $G$, we can transform $\alpha$ into $\beta$ by recoloring each vertex at most $c$ times, where $c = c(d)$ is a fixed constant only depending on $d$.

Figures (2)

  • Figure 1: A sample of the pattern in the proof of statement 2. The blue numbers represent colors.
  • Figure 2: A sample of the pattern in the proof of statement 3. The blue numbers represent colors.

Theorems & Definitions (9)

  • Conjecture 1.1: Cereceda cereceda2007mixing, Conjecture 5.21
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1
  • Proposition 4.1: bartier2021recoloring
  • Lemma 4.4
  • Lemma 4.5
  • Lemma 4.6
  • Claim 4.7