On strong incidence coloring of subcubic graphs

Abstract

An incidence of a graph [Formula: see text] is a pair [Formula: see text] where [Formula: see text] is a vertex of [Formula: see text] and [Formula: see text] is an edge of [Formula: see text] incident with [Formula: see text]. Two incidences [Formula: see text] and [Formula: see text] of [Formula: see text] are adjacent whenever (i) [Formula: see text], or (ii) [Formula: see text] or (iii) [Formula: see text] or [Formula: see text]. A strong incidence coloring of a graph [Formula: see text] is a mapping from the set of incidences of [Formula: see text] to the set of colors [Formula: see text], such that every two incidences that are adjacent, or adjacent to the same incidence receive distinct colors. In this paper, we prove that every connected subcubic graph [Formula: see text] except [Formula: see text] has a strong incidence coloring with at most [Formula: see text] colors.