Hadwiger's Conjecture with Certain Forbidden Induced Subgraphs

TL;DR

This paper proves that certain classes of graphs, specifically those free of specific small induced subgraphs, always satisfy Hadwiger's Conjecture, with most proofs supported by computer assistance.

Contribution

It extends previous results by showing that graphs avoiding certain small induced subgraphs are not counterexamples to Hadwiger's Conjecture, including 33 specific graphs and K8.

Findings

Graphs avoiding 3 and H are not counterexamples to Hadwiger's Conjecture.

Most proofs are computer-assisted, confirming the conjecture for these classes.

The result improves upon earlier partial results by multiple researchers.

Abstract

We prove that $\{\overline{K_3}, H\}$-free graphs are not counterexamples to Hadwiger's Conjecture, where $H$ is any one of 33 graphs on seven, eight, or nine vertices, or $H=K_8$. This improves on past results of Plummer-Stiebitz-Toft, Kriesell, and Bosse. The proofs are mostly computer-assisted.