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On the finiteness of $k$-vertex-critical $2P_2$-free graphs with forbidden induced squids or bulls

Melvin Adekanye, Christopher Bury, Ben Cameron, Thaler Knodel

TL;DR

The paper proves that for all $k$, there are finitely many $k$-vertex-critical $(2P_2,H)$-free graphs when $H$ is one of $bull$, $chair$, $claw+P_1$, or $\overline{diamond+P_1}$, by reducing to $(P_3+mP_1)$-free structures and applying a known finiteness result for such classes. It further derives finite-vertex-critical results for $(2P_2,(4,\ell)$-$squid$)-free and $(2P_2,(3,\ell)$-$squid$)-free graphs (with $m=4,3$ respectively), yielding corollaries for the corresponding order-$5$ forbidden subgraphs. The authors also provide computer-assisted exhaustive generation of all $k$-vertex-critical $(2P_2,H)$-free graphs for $H=bull$ or $H=banner$ with $k\le 7$, enabling certifying $k$-coloring in these families. Overall, the work advances the understanding of finiteness and certifying colorability in restricted graph classes and introduces a broadly applicable reduction technique to $(P_3+mP_1)$-free graphs.” wrapped in $...$ where appropriate.}

Abstract

A graph is $k$-vertex-critical if $χ(G)=k$ but $χ(G-v)<k$ for all $v\in V(G)$ and $(G,H)$-free if it contains no induced subgraph isomorphic to $G$ or $H$. We show that there are only finitely many $k$-vertex-critical $(2P_2,H)$-free graphs for all $k$ when $H$ is isomorphic to any of the following graphs of order $5$: $bull$, $chair$, $claw+P_1$, or $\overline{diamond+P_1}$. The latter three are corollaries of more general results where $H$ is isomorphic to $(m, \ell)$-$squid$ for $m=3,4$ and any $\ell\ge 1$ where an $(m,\ell)$-$squid$ is the graph obtained from an $m$-cycle by attaching $\ell$ leaves to a single vertex of the cycle. For each of the graphs $H$ above and any fixed $k$, our results imply the existence of polynomial-time certifying algorithms for deciding the $k$-colourability problem for $(2P_2,H)$-free graphs. Further, our structural classifications allow us to exhaustively generate, with aid of computer search, all $k$-vertex-critical $(2P_2,H)$-free graphs for $k\le 7$ when $H=bull$ or $H=(4,1)$-$squid$ (also known as $banner$).

On the finiteness of $k$-vertex-critical $2P_2$-free graphs with forbidden induced squids or bulls

TL;DR

The paper proves that for all , there are finitely many -vertex-critical -free graphs when is one of , , , or , by reducing to -free structures and applying a known finiteness result for such classes. It further derives finite-vertex-critical results for -)-free and -)-free graphs (with respectively), yielding corollaries for the corresponding order- forbidden subgraphs. The authors also provide computer-assisted exhaustive generation of all -vertex-critical -free graphs for or with , enabling certifying -coloring in these families. Overall, the work advances the understanding of finiteness and certifying colorability in restricted graph classes and introduces a broadly applicable reduction technique to -free graphs.” wrapped in where appropriate.}

Abstract

A graph is -vertex-critical if but for all and -free if it contains no induced subgraph isomorphic to or . We show that there are only finitely many -vertex-critical -free graphs for all when is isomorphic to any of the following graphs of order : , , , or . The latter three are corollaries of more general results where is isomorphic to - for and any where an - is the graph obtained from an -cycle by attaching leaves to a single vertex of the cycle. For each of the graphs above and any fixed , our results imply the existence of polynomial-time certifying algorithms for deciding the -colourability problem for -free graphs. Further, our structural classifications allow us to exhaustively generate, with aid of computer search, all -vertex-critical -free graphs for when or - (also known as ).
Paper Structure (9 sections, 14 theorems, 3 figures, 1 table)

This paper contains 9 sections, 14 theorems, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Outline
  3. Notation
  4. Preliminaries
  5. ($2P_2$, $bull$)-free
  6. ($2P_2$, $(4,\ell)$-$squid$)-free
  7. ($2P_2$, $(3,\ell)$-$squid$)-free
  8. Exhaustive Generation for Small $k$
  9. Conclusion

Key Result

Lemma 2.1

Let $G$ be a graph with chromatic number $k$. If G contains two disjoint $m$-cliques $A = \{a_1, a_2,\ldots , a_m\}$ and $B = \{b_1, b_2,\ldots , b_m\}$ such that $N(a_i) \setminus A \subseteq N(b_i) \setminus B$ for all $1 \le i \le m$, then $G$ is not $k$-vertex-critical.

Figures (3)

  • Figure 1: The general form of the $(m,\ell)$-$squid$ graphs for $m=3$ and $m=4$.
  • Figure 2: The $bull$ graph.
  • Figure 3: An illustration of part of the proof of Lemma \ref{['lem:2P2bullP3P1fee']} with the induced $bull$ in bold.

Theorems & Definitions (19)

  • Lemma 2.1: Hoang2015
  • Theorem 2.2: CameronHoangSawada2022
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • Lemma 4.1
  • ...and 9 more