Incidence coloring of Mycielskians with fast algorithm

Abstract

An incidence of a graph G is a vertex-edge pair (v,e) such that the vertex v is incident with the edge e. A proper incidence k-coloring of a graph is a coloring of its incidences involving k colors so that two incidences (u,e) and (w,f) receive distinct colors if and only if u=w, or e=f, or uw∈{e,f}. In this paper, we present some idea of using the incidence coloring to model a kind of multi-frequency assignment problem, in which each transceiver can be simultaneously in both sending and receiving modes, and then establish some theoretical and algorithmic aspects of the incidence coloring.Specifically, we conjecture that if G is the Mycielskian of some graph then it has a proper incidence (Δ(G)+2)-coloring. Actually, our conjecture is motivated by the “(Δ+2) conjecture” of Brualdi and Quinn Massey in 1993, which states that every graph G has a proper incidence (Δ(G)+2)-coloring, and was disproved in 1997 by Guiduli, who pointed out that the Paley graphs with large maximum degree are counterexamples (yet they are all known counterexamples to the “(Δ+2) conjecture”, and are not Mycielskians of any graph).To support our conjecture, we prove in this paper that if G is the Mycielskian of a graph H with |H|≥3Δ(H)+2, then we can construct a proper incidence (Δ(G)+1)-coloring of G in cubic time, and if G is the Mycielskian of an incidence (Δ(H)+1)-colorable graph H with |H|≤2Δ(H), or the Mycielskian of an incidence (Δ(H)+2)-colorable graph H with |H|≥2Δ(H)+1, then G has a proper incidence (Δ(G)+2)-coloring. The minimum positive integer k such that the Mycielskian of a cycle or a path has a proper incidence k-coloring is also determined.