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Dominating Hadwiger's Conjecture holds for all $2K_2$-free graphs

Zi-Xia Song, Thomas Tibbetts

TL;DR

The paper addresses whether every graph with chromatic number $t$ contains a dominating $K_t$ minor and proves this stronger property for the class of $2K_2$-free graphs, showing $h_d(G) \geq \chi(G)$. The authors leverage a novel structural component—an induced banner formed by adjoining a vertex to exactly one vertex of a $C_4$—together with a key forbidhole-based lemma to guide a minimum-counterexample analysis. The main contribution is a complete proof that every $2K_2$-free graph contains a dominating $K_{\chi(G)}$ minor, advancing the understanding of the Dominating Hadwiger Conjecture within a broad graph class. This result also informs the broader interplay between graph coloring, minors, and induced subgraph structures, and may influence future investigations of the conjecture's limits and extensions.

Abstract

A dominating $K_t$ minor in a graph $G$ is a sequence $(T_1,\dots,T_t)$ of pairwise disjoint non-empty connected subgraphs of $G$, such that for $1 \leq i<j\leq t$, every vertex in $T_j$ has a neighbor in $T_i$. Replacing ``every vertex in $T_j$'' by ``some vertex in $T_j$'' retrieves the standard definition of a $K_t$ minor. The strengthened notion was introduced by Illingworth and Wood [arXiv:2405.14299], who asked whether every graph with chromatic number $t$ contains a dominating $K_t$ minor. This is a substantial strengthening of the celebrated Hadwiger's Conjecture, which asserts that every graph with chromatic number $t$ contains a $K_t$ minor. At the ``New Perspectives in Colouring and Structure'' workshop held at the Banff International Research Station from September 29 - October 4, 2024, Norin referred to this question as the ``Dominating Hadwiger's Conjecture'' and believes it is likely false. In this paper we prove that the Dominating Hadwiger's Conjecture holds for all $2K_2$-free graphs. A key component of our proof is the clever use of the existence of an induced banner, obtained by adding a vertex adjacent to exactly one vertex on a cycle of length four.

Dominating Hadwiger's Conjecture holds for all $2K_2$-free graphs

TL;DR

The paper addresses whether every graph with chromatic number contains a dominating minor and proves this stronger property for the class of -free graphs, showing . The authors leverage a novel structural component—an induced banner formed by adjoining a vertex to exactly one vertex of a —together with a key forbidhole-based lemma to guide a minimum-counterexample analysis. The main contribution is a complete proof that every -free graph contains a dominating minor, advancing the understanding of the Dominating Hadwiger Conjecture within a broad graph class. This result also informs the broader interplay between graph coloring, minors, and induced subgraph structures, and may influence future investigations of the conjecture's limits and extensions.

Abstract

A dominating minor in a graph is a sequence of pairwise disjoint non-empty connected subgraphs of , such that for , every vertex in has a neighbor in . Replacing ``every vertex in '' by ``some vertex in '' retrieves the standard definition of a minor. The strengthened notion was introduced by Illingworth and Wood [arXiv:2405.14299], who asked whether every graph with chromatic number contains a dominating minor. This is a substantial strengthening of the celebrated Hadwiger's Conjecture, which asserts that every graph with chromatic number contains a minor. At the ``New Perspectives in Colouring and Structure'' workshop held at the Banff International Research Station from September 29 - October 4, 2024, Norin referred to this question as the ``Dominating Hadwiger's Conjecture'' and believes it is likely false. In this paper we prove that the Dominating Hadwiger's Conjecture holds for all -free graphs. A key component of our proof is the clever use of the existence of an induced banner, obtained by adding a vertex adjacent to exactly one vertex on a cycle of length four.
Paper Structure (2 sections, 5 theorems, 12 equations, 1 figure)

This paper contains 2 sections, 5 theorems, 12 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['t:main']}

Key Result

Theorem 1.3

Every $2K_2$-free graph $G$ satisfies $h_d(G) \geq \chi(G)$.

Figures (1)

  • Figure 1: The graph $T$ in which $T[\{b_1, b_2, b_3, b, b'\}]=B$ is a banner.

Theorems & Definitions (18)

  • Conjecture 1.1: Hadwiger's Conjecture Had43
  • Conjecture 1.2: Dominating Hadwiger's Conjecture IW24
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5: Scully and Song SS25
  • Theorem 1.6: Micu Micu05
  • proof
  • Theorem 2.1
  • proof
  • proof
  • ...and 8 more