Strengthening Hadwiger's conjecture for $4$- and $5$-chromatic graphs

TL;DR

This paper proves a strengthened version of Hadwiger's conjecture for 4-chromatic graphs and derives a stronger result for 5-chromatic graphs, showing the existence of specific minors rooted at certain vertex sets.

Contribution

It confirms Holroyd's conjecture for t=4 and establishes a stronger form of Hadwiger's conjecture for t=5, with explicit root-set conditions.

Findings

Proves Holroyd's conjecture for t=4.

Shows every 5-chromatic graph contains a K_5-minor with a singleton branch-set.

In 5-vertex-critical graphs, the singleton branch-set can be any vertex.

Abstract

Hadwiger's famous coloring conjecture states that every $t$-chromatic graph contains a $K_t$-minor. Holroyd [Bull. London Math. Soc. 29, (1997), pp. 139--144] conjectured the following strengthening of Hadwiger's conjecture: If $G$ is a $t$-chromatic graph and $S \subseteq V(G)$ takes all colors in every $t$-coloring of $G$, then $G$ contains a $K_t$-minor rooted at $S$. We prove this conjecture in the first open case of $t=4$. Notably, our result also directly implies a stronger version of Hadwiger's conjecture for $5$-chromatic graphs as follows: Every $5$-chromatic graph contains a $K_5$-minor with a singleton branch-set. In fact, in a $5$-vertex-critical graph we may specify the singleton branch-set to be any vertex of the graph.