Incidence Choosability of Graphs

TL;DR

This paper explores the list version of incidence colouring in graphs, establishing exact or bounded incidence choice numbers for various graph classes, advancing understanding of graph colouring complexities.

Contribution

It introduces the list incidence colouring concept and determines the incidence choice number for specific graph classes, a novel extension of prior incidence colouring studies.

Findings

Exact incidence choice number for square grids

Upper bounds for Halin graphs and cactuses

Results for Hamiltonian cubic graphs

Abstract

An incidence of a graph G is a pair (v, e) where v is a vertex of G and e is an edge of G incident with v. Two incidences (v, e) and (w, f) of G are adjacent whenever (i) v = w, or (ii) e = f , or (iii) vw = e or f. An incidence p-colouring of G is a mapping from the set of incidences of G to the set of colours {1,. .. , p} such that every two adjacent incidences receive distinct colours. Incidence colouring has been introduced by Brualdi and Quinn Massey in 1993 and, since then, studied by several authors. In this paper, we introduce and study the list version of incidence colouring. We determine the exact value of -- or upper bounds on -- the incidence choice number of several classes of graphs, namely square grids, Halin graphs, cactuses and Hamiltonian cubic graphs.