More results on the z-chromatic number of graphs

Abstract

By a z-coloring of a graph G we mean any proper vertex coloring consisting of the color classes C1,…,Ck such that (i) for any two colors i and j with 1≤i<j≤k, any vertex of color j is adjacent to a vertex of color i, (ii) there exists a set {u1,…,uk} of vertices of G such that uj∈Cj for any j∈{1,…,k}; also for any j≠k with 1≤j<k, vertex uk is adjacent to uj and (iii) for each i and j with i≠j, the vertex uj has a neighbor in Ci. Denote by z(G) the maximum number of colors used in any z-coloring of G. Denote the Grundy and b-chromatic numbers of G by Γ(G) and b(G), respectively. The z-coloring is an improvement over both the Grundy and b-coloring of graphs. We prove that z(G) is much better than min{Γ(G),b(G)} for infinitely many graphs G by obtaining an infinite sequence {Gn}n=3∞ of graphs such that z(Gn)=n but Γ(Gn)=b(Gn)=2n−1 for each n≥3. We show that acyclic graphs are z-monotonic and z-continuous. Then it is proved that to decide whether z(G)=Δ(G)+1 is NP-complete even for bipartite graphs G. We finally prove that to recognize graphs G satisfying z(G)=χ(G) is coNP-complete, improving a previous result for the Grundy number.