Coloring graphs without fan vertex-minors and graphs without cycle pivot-minors
Ilkyoo Choi, O-joung Kwon, Sang-il Oum

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