Every graph with no $K_7^{\vee}$-minor is $6$-colorable

TL;DR

This paper proves that any graph excluding a specific minor, $K_7^{ vee}$, can be colored with six colors, advancing understanding related to Hadwiger's conjecture.

Contribution

It establishes that graphs without a $K_7^{ vee}$-minor are 6-colorable, providing a new partial result towards Hadwiger's conjecture.

Findings

Graphs with no $K_7^{ vee}$-minor are 6-colorable

Supports Hadwiger's conjecture for this class of graphs

Advances minor exclusion coloring theory

Abstract

Let $K_7^{\vee}$ denote the graph obtained from the complete graph on seven vertices by deleting two edges with a common end. Motivated by Hadwiger's conjecture, we prove that every graph with no $K_7^{\vee}$-minor is $6$-colorable.