Coloring graphs without fan vertex-minors and graphs without cycle pivot-minors

TL;DR

This paper proves that graphs with sufficiently large chromatic number necessarily contain either large cliques or specific vertex- or pivot-minors, extending understanding of graph coloring and minor containment.

Contribution

It establishes new results linking large chromatic number to the presence of fan and cycle pivot-minors, broadening the theory of graph minors and coloring.

Findings

Graphs with large chromatic number contain fan vertex-minors.

Graphs with large chromatic number contain cycle pivot-minors.

Results apply to all positive integers q and k under certain conditions.

Abstract

A fan $F_k$ is a graph that consists of an induced path on $k$ vertices and an additional vertex that is adjacent to all vertices of the path. We prove that for all positive integers $q$ and $k$, every graph with sufficiently large chromatic number contains either a clique of size $q$ or a vertex-minor isomorphic to $F_k$. We also prove that for all positive integers $q$ and $k\ge 3$, every graph with sufficiently large chromatic number contains either a clique of size $q$ or a pivot-minor isomorphic to a cycle of length $k$.