General Parity Result and Cycle-plus-Triangles Graphs

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.