Answers to some problems about graph coloring test graphs

TL;DR

This paper investigates properties of graph coloring test graphs, proving certain graphs are homotopy test graphs and exploring conditions under which graphs are Stiefel-Whitney test graphs, addressing open problems in the field.

Contribution

It proves that graphs with chromatic number 2 are homotopy test graphs and constructs a graph with two involutions where only one yields a Stiefel-Whitney test graph.

Findings

Graphs with chromatic number 2 are homotopy test graphs

Existence of a graph with two involutions where only one is a Stiefel-Whitney test graph

Answers to open problems posed by Kozlov

Abstract

We prove that a graph whose chromatic number is 2 is a homotopy test graph. We also prove that there is a graph $K$ with two involutions $\gamma_1$ and $\gamma_2$ such that $(K,\gamma_1)$ is a Stiefel-Whitney test graph, but $(K,\gamma_2)$ is not. These are answers to some of the problems suggested by Kozlov.