Proof of the Clustered Hadwiger Conjecture

TL;DR

This paper proves that graphs excluding certain minors can be coloured with a fixed number of colours such that monochromatic components are bounded in size, solving longstanding open problems and extending Hadwiger's Conjecture.

Contribution

It establishes the first improper analogue of Hadwiger's Conjecture with optimal colour bounds and bounded monochromatic components for various classes of minor-free graphs.

Findings

Proves that $K_h$-minor-free graphs are $(h-1)$-colourable with bounded monochromatic components.

Shows that $K_{s,t}$-minor-free graphs are $(s+1)$-colourable with bounded monochromatic components.

Extends results to $X$-minor-free graphs excluding apex minors, with optimal colour bounds.

Abstract

Hadwiger's Conjecture asserts that every $K_h$-minor-free graph is properly $(h-1)$-colourable. We prove the following improper analogue of Hadwiger's Conjecture: for fixed $h$, every $K_h$-minor-free graph is $(h-1)$-colourable with monochromatic components of bounded size. The number of colours is best possible regardless of the size of monochromatic components. It solves an open problem of Edwards, Kang, Kim, Oum and Seymour [\emph{SIAM J. Disc. Math.} 2015], and concludes a line of research initiated in 2007. Similarly, for fixed $t\geq s$, we show that every $K_{s,t}$-minor-free graph is $(s+1)$-colourable with monochromatic components of bounded size. The number of colours is best possible, solving an open problem of van de Heuvel and Wood [\emph{J.~London Math.\ Soc.} 2018]. We actually prove a single theorem from which both of the above results are immediate corollaries. For an excluded apex minor, we strengthen the result as follows: for fixed $t\geq s\geq 3$, and for any fixed apex graph $X$, every $K_{s,t}$-subgraph-free $X$-minor-free graph is $(s+1)$-colourable with monochromatic components of bounded size. The number of colours is again best possible.