Every graph with no $\mathcal{K}_9^{-6}$ minor is $8$-colorable

TL;DR

This paper proves that graphs excluding a specific minor related to $K_9$ are 8-colorable, extending previous results and supporting a broader conjecture linking minors and chromatic number.

Contribution

It establishes that graphs with no $ ext{K}_9^{-6}$ minor are 8-colorable, advancing the understanding of graph coloring in relation to minor exclusion.

Findings

Graphs with no $ ext{K}_9^{-6}$ minor are 8-colorable

Supports the $H$-Hadwiger's Conjecture for certain nine-vertex graphs

Extends previous minor exclusion coloring bounds

Abstract

For positive integers $t$ and $s$, let $\mathcal{K}_t^{-s}$ denote the family of graphs obtained from the complete graph $K_t$ by removing $s$ edges. A graph $G$ has no $\mathcal{K}_t^{-s}$ minor if it has no $H$ minor for every $H\in \mathcal{K}_t^{-s}$. Motivated by the famous Hadwiger's Conjecture, Jakobsen in 1971 proved that every graph with no $\mathcal{K}_7^{-2}$ minor is $6$-colorable; very recently the present authors proved that every graph with no $\mathcal{K}_8^{-4}$ minor is $7$-colorable. In this paper we continue our work and prove that every graph with no $\mathcal{K}_9^{-6}$ minor is $8$-colorable. Our result implies that $H$-Hadwiger's Conjecture, suggested by Paul Seymour in 2017, is true for all graphs $H$ on nine vertices such that $H$ is a subgraph of every graph in $ \mathcal{K}_9^{-6}$.