Incidence coloring of k-degenerated graphs

Incidence Coloring Game and Arboricity of Graphs

An incidence of a graph G is a pair (v, e) where v is a vertex of G and e an edge incident to v. Two incidences (v, e) and (w, f) are adjacent whenever v = w, or e = f , or vw = e or f . The incidence coloring game [S.D. Andres, The incidence game chromatic number, Discrete Appl. Math. 157 (2009), 1980–1987] is a variation of the ordinary coloring game where the two players, Alice and Bob, alte...

Interval incidence coloring of subcubic graphs

For a given simple graph G = (V,E), we define an incidence as a pair (v, e), where vertex v ∈ V (G) is one of the ends of edge e ∈ E(G). Let us define a set of incidences I(G) = {(v, e) : v ∈ V (G)∧ e ∈ E(G)∧ v ∈ e}. We say that two incidences (v, e) and (w, f) are adjacent if one of the following holds: (i) v = w, e 6= f , (ii) e = f , v 6= w, (iii) e = {v, w}, f = {w, u} and v 6= u. By an inc...

The d-relaxed game chromatic index of k-degenerated graphs

(k, 1)-coloring of Sparse Graphs

A graph G is called (k, 1)-colorable, if the vertex set of G can be partitioned into subsets V1 and V2 such that the graph G[V1] induced by the vertices of V1 has maximum degree at most k and the graph G[V2] induced by the vertices of V2 has maximum degree at most 1. We prove that every graph with a maximum average degree less than 10k+22 3k+9 admits a (k, 1)-coloring, where k ≥ 2. In particula...

On incidence coloring for some cubic graphs

In 1993, Brualdi and Massey conjectured that every graph can be incidence colored with ∆+2 colors, where ∆ is the maximum degree of a graph. Although this conjecture was solved in the negative by an example in [1], it might hold for some special classes of graphs. In this paper, we consider graphs with maximum degree ∆ = 3 and show that the conjecture holds for cubic Hamiltonian graphs and some...