Incidence coloring - Cold cases

František Kardoš , Mária Maceková (1) , Martina Mockovčiaková (2) , Eric Sopena (3) , Roman Soták (1)

1 Institute of Mathematics [Kosice, Slovakia]
2 University of West Bohemia - Department of Mathematics
3 LaBRI - Laboratoire Bordelais de Recherche en Informatique

An incidence in a graph G is a pair (v, e) where v is a vertex of G and e is an edge of G incident to v. Two incidences (v, e) and (u, f) are adjacent if at least one of the following holds: (i) v = u, (ii) e = f, or (iii) edge vu is from the set {e, f}. An incidence coloring of G is a coloring of its incidences assigning distinct colors to adjacent incidences. The minimum number of colors needed for incidence coloring of a graph is called the incidence chromatic number. It was proved that at most Delta(G) + 5 colors are enough for an incidence coloring of any planar graph G except for Delta(G) = 6, in which case at most 12 colors are needed. It is also known that every planar graph G with girth at least 6 and Delta(G) >= 5 has incidence chromatic number at most Delta(G)+ 2. In this paper we present some results on graphs regarding their maximum degree and maximum average degree. We improve the bound for planar graphs with Delta(G) = 6. We show that the incidence chromatic number is at most Delta(G) + 2 for any graph G with mad(G) < 3 and Delta(G) = 4, and for any graph with mad(G) < 10/3 and Delta(G) >= 8.

Domaines: Mathématique discrète [cs.DM]

František Kardoš, Mária Maceková, Martina Mockovčiaková, Eric Sopena, Roman Soták. Incidence coloring - Cold cases. Discussiones Mathematicae Graph Theory, 2020, 40 (1), pp.345-354. ⟨10.7151/dmgt.2140⟩. ⟨hal-02264649⟩