Graphs with no $K_9^=$ minor are 10-colorable

TL;DR

This paper proves that graphs lacking a $K_9^=$ minor can be colored with at most 10 colors, extending previous results for smaller t and contributing to the understanding of graph coloring related to minor exclusions.

Contribution

The paper establishes that graphs with no $K_9^=$ minor are 10-colorable, extending known bounds from smaller t values.

Findings

Graphs with no $K_9^=$ minor are 10-colorable.

Extends previous results for $K_t^=$ minors with t ≤ 8.

Contributes to the study of graph coloring and minor exclusion conjectures.

Abstract

Hadwiger's conjecture claims that any graph with no $K_t$ minor is $(t - 1)$-colorable. This has been proved for $t \le 6$, but remains open for $t \ge 7$. As a variant of this conjecture, graphs with no $K_t^=$ minor have been considered, where $K_t^=$ denotes the complete graph with two edges removed. It has been shown that graphs with no $K_t^=$ minor are $(2t - 8)$-colorable for $t \in \{7, 8\}$. In this paper, we extend this result to the case $t = 9$ and show that graphs with no $K_9^=$ minor are $10$-colorable.