Incidence choosability of 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.

Strong incidence colouring of graphs

“…For any graph G, the incidence graph of G, denoted by I G , introduced in [1], is the graph whose vertices are the incidences of G, two incidences being joined by an edge whenever they are adjacent. Clearly, every incidence colouring of G is nothing but a proper vertex colouring of I G , and every strong incidence colouring of G is nothing but a 2-distance colouring of I G , so χ i (G) = χ(I G ) and χ s i (G) = χ 2 (I G ).…”

On incidence choosability of cubic graphs

“…The first lemma easily follows from the fact that each incidence of a subcubic graph G has at most 7 forbidden colors from its adjacent incidences. We can extend φ by choosing coloring for the incidences on e. The following lemma was presented in [3], and we write its proof here for the sake of completeness.…”

“…The incidence choice number ch i (G) of a multigraph G is the smallest integer k such that G is incidence k-choosable. Whereas the incidence chromatic number for a cubic graph is known to be at most 5 as in Theorem 1.1, the incidence choice number of a cubic graph is firstly studied in [3] most recently as follows.…”

Defective incidence coloring of graphs

“…There are many interesting variations of incidence coloring of graphs, including incidence list coloring [5], incidence game coloring [2,3], interval incidence coloring [11], fractional incidence coloring [17], and oriented incidence coloring [9].…”