General Parity Result and Cycle‐Plus‐Triangles Graphs

Abstract

We generalize a parity result of Fleishner and Stiebitz that being combined with AlonTarsi 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.We start with the following parity theorem. Theorem 1. Let V = ⊔ n i=1 V i be a finite set partitioned onto disjoint subsets V i of odd sizes |V i |. Let G be a graph on a ground set V such that each V i is independent set in G and each bipartite subgraph induced on V i ⊔ V j is Eulerian (i.e. all degrees are even). Consider the subsets U ⊂ V such that |U ∩ V i | = 1 for all i and subgraph induced on U is Eulerian. Then the number of such U is odd.