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Limits of degeneracy for colouring graphs with forbidden minors

Sergey Norin, Jérémie Turcotte

TL;DR

This work addresses Seymour's relaxation of Hadwiger's conjecture by proving that every sufficiently large bipartite graph $H$ in any monotone family with strongly sublinear separators, and with bounded maximum degree, is Hadwiger-amenable: any non-null graph $G$ with $\delta(G)\ge v(H)-1$ contains $H$ as a minor. The authors develop a dense-pair framework and a multi-faceted toolbox, combining models of minors, extremal-density bounds for bipartite graphs, density-increment arguments, and a novel minor-from-pieces construction to assemble a model of $H$ inside $G$ from dense components. They organize the argument into a three-case plan (small, intermediate, and large graphs) and prove a sequence of auxiliary results, including a small-case embedding lemma, a bounded-component minor theorem, and a density-increment machinery that yields minors from dense fragments. The paper also establishes tightness, showing that each assumption is essential by constructing counterexamples if any condition is dropped. Overall, the results extend Hadwiger-type degeneracy bounds to broad classes of $H$ and highlight a density-based route to discovering large bipartite minors within dense graphs, with potential implications for coloring with forbidden minors and related structural graph theory questions.

Abstract

Motivated by Hadwiger's conjecture, Seymour asked which graphs $H$ have the property that every non-null graph $G$ with no $H$ minor has a vertex of degree at most $|V(H)|-2$. We show that for every monotone graph family $\mathcal{F}$ with strongly sublinear separators, all sufficiently large bipartite graphs $H \in \mathcal{F}$ with bounded maximum degree have this property. None of the conditions that $H$ belongs to $\mathcal{F}$, that $H$ is bipartite and that $H$ has bounded maximum degree can be omitted.

Limits of degeneracy for colouring graphs with forbidden minors

TL;DR

This work addresses Seymour's relaxation of Hadwiger's conjecture by proving that every sufficiently large bipartite graph in any monotone family with strongly sublinear separators, and with bounded maximum degree, is Hadwiger-amenable: any non-null graph with contains as a minor. The authors develop a dense-pair framework and a multi-faceted toolbox, combining models of minors, extremal-density bounds for bipartite graphs, density-increment arguments, and a novel minor-from-pieces construction to assemble a model of inside from dense components. They organize the argument into a three-case plan (small, intermediate, and large graphs) and prove a sequence of auxiliary results, including a small-case embedding lemma, a bounded-component minor theorem, and a density-increment machinery that yields minors from dense fragments. The paper also establishes tightness, showing that each assumption is essential by constructing counterexamples if any condition is dropped. Overall, the results extend Hadwiger-type degeneracy bounds to broad classes of and highlight a density-based route to discovering large bipartite minors within dense graphs, with potential implications for coloring with forbidden minors and related structural graph theory questions.

Abstract

Motivated by Hadwiger's conjecture, Seymour asked which graphs have the property that every non-null graph with no minor has a vertex of degree at most . We show that for every monotone graph family with strongly sublinear separators, all sufficiently large bipartite graphs with bounded maximum degree have this property. None of the conditions that belongs to , that is bipartite and that has bounded maximum degree can be omitted.
Paper Structure (10 sections, 30 theorems, 64 equations)

This paper contains 10 sections, 30 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Proof outline
  3. Tools
  4. Proof outline
  5. Small case
  6. Minors with bounded component size
  7. Density increment
  8. Building a minor from pieces
  9. Tightness
  10. Proof of \ref{['lem:compsize']}

Key Result

Theorem 1.4

For every graph family $\mathcal{F}$ with strongly sublinear separators and every $\Delta \in \mathbb{N}$, there exists $M=M_{thm:main}(\mathcal{F},\Delta)$ satisfying the following. If $H\in \mathcal{F}$ is a bipartite graph with $\Delta(H)\leq \Delta$ and $\textup{v}(H)\geq M$ then $H$ is Hadwiger

Theorems & Definitions (48)

  • Conjecture 1.1: Hadwiger's conjecture hadwiger_uber_1943
  • Conjecture 1.2: Seymour nash_hadwigers_2016seymour_birs_2017
  • Theorem 1.4
  • Lemma 2.1
  • Theorem 2.2: haslegrave_extremal_2022, hendrey_extremal_2022
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4: hendrey_extremal_2022
  • Theorem 2.5
  • ...and 38 more