General Parity Result and Cycle-plus-Triangles Graphs

TL;DR

This paper extends a parity result and applies a modified polynomial method to analyze 3-choosability in graphs formed by Hamiltonian cycles and disjoint triangles, providing new combinatorial insights.

Contribution

It generalizes a parity theorem and introduces a modified polynomial approach that yields more combinatorial information about graph colorings.

Findings

A generalized parity result for specific graph classes.

A modified polynomial method that enhances combinatorial analysis.

Proof that certain 4-regular graphs are 3-choosable.

Abstract

We generalize a parity result of Fleishner and Stiebitz that being combined with Alon--Tarsi polynomial method allowed them to prove that a 4-regular graph formed by a Hamiltonian cycle and several disjoint triangles is always 3-choosable. Also we present a modification of polynomial method and show how it gives slightly more combinatorial information about colourings than direct application of Alon's Combinatorial Nullstellensatz.