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On the structures of {diamond, bowtie}-free graphs that do not contain an induced subdivision of $K_4$

Feng Liu, Shuang Sun, Yan Wang

Abstract

A graph is $\mathrm{ISK}_4$-free if it contains no induced subdivision of $K_4$. Lévêque et al. [\emph{J. Combin. Theory Ser. B} \textbf{102} (2012) 924--947] conjectured that all $\mathrm{ISK}_4$-free graphs are 4-colorable. Chen et al. [\emph{J. Graph Theory} \textbf{96} (2021) 554--577] proved that $\{\mathrm{ISK}_4, \mathrm{diamond}, \mathrm{bowtie}\}$-free graphs are 4-colorable and asked whether such graphs are 3-colorable, where a diamond is $K_4$ minus one edge and a bowtie consists of two triangles sharing a vertex. In this paper, we characterize the structures of $\{\mathrm{ISK}_4, \mathrm{diamond}, \mathrm{bowtie}\}$-free graphs and prove that such graphs are 3-colorable, which answers a question of Chen et al. [\emph{J. Graph Theory} \textbf{96} (2021) 554--577] affirmatively and extends a result of Chudnovsky et al. [\emph{J. Graph Theory} \textbf{92} (2019) 67--95]. Furthermore, our structural theorem yields a polynomial-time algorithm for decomposing $\{\mathrm{ISK}_4, \mathrm{diamond}, \mathrm{bowtie}\}$-free graphs, and consequently a polynomial-time algorithm for coloring this class of graphs.

On the structures of {diamond, bowtie}-free graphs that do not contain an induced subdivision of $K_4$

Abstract

A graph is -free if it contains no induced subdivision of . Lévêque et al. [\emph{J. Combin. Theory Ser. B} \textbf{102} (2012) 924--947] conjectured that all -free graphs are 4-colorable. Chen et al. [\emph{J. Graph Theory} \textbf{96} (2021) 554--577] proved that -free graphs are 4-colorable and asked whether such graphs are 3-colorable, where a diamond is minus one edge and a bowtie consists of two triangles sharing a vertex. In this paper, we characterize the structures of -free graphs and prove that such graphs are 3-colorable, which answers a question of Chen et al. [\emph{J. Graph Theory} \textbf{96} (2021) 554--577] affirmatively and extends a result of Chudnovsky et al. [\emph{J. Graph Theory} \textbf{92} (2019) 67--95]. Furthermore, our structural theorem yields a polynomial-time algorithm for decomposing -free graphs, and consequently a polynomial-time algorithm for coloring this class of graphs.
Paper Structure (7 sections, 45 theorems, 5 equations, 23 figures, 4 algorithms)

This paper contains 7 sections, 45 theorems, 5 equations, 23 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Proper wheels
  4. Proper wheel centers
  5. Properties of $\left\{\textnormal{ISK}_4, \textnormal{triangle}, K_{3,3}\right\}$-free graphs
  6. Main result
  7. Decomposition and coloring algorithm for $\left\{\textnormal{ISK}_4, \textnormal{diamond}, \textnormal{bowtie}\right\}$-free graphs

Key Result

Theorem 1.1

Let $G$ be an $\textnormal{ISK}_4$-free graph. Then either $G$ is series-parallel, or $G$ is the line graph of a graph with maximum degree at most three, or $G$ has a clique-cutset, a proper $2$-cutset, a star-cutset, or a double star-cutset.

Figures (23)

  • Figure 1: A diamond and a bowtie.
  • Figure 2: Illustration of a prism.
  • Figure 3: The neighbors of $a$, $b$, and $c$ on the hole $C$ and their locations on $C$.
  • Figure 4: Illustrations of the configuration: a tripod graph.
  • Figure 5: Illustrations of the possible scenario of two vertex-disjoint paths $P_1$ and $P_2$ in $T$, highlighted in red.
  • ...and 18 more figures

Theorems & Definitions (49)

  • Theorem 1.1: Lévêque et al. Leveque2012
  • Conjecture 1.2: Lévêque et al. Leveque2012
  • Theorem 1.3: Chudnovsky et al. Chudnovsky2019
  • Theorem 1.4: Chudnovsky et al. Chudnovsky2019
  • Theorem 1.5: Chen et al. Chen2021
  • Theorem 1.7
  • Theorem 1.8
  • Lemma 2.1: Lévêque et al. Leveque2012
  • Theorem 2.2: Lévêque et al. Leveque2012
  • Lemma 2.3: Chudnovsky et al. Chudnovsky2019
  • ...and 39 more