# More results on the z-chromatic number of graphs

By a z-coloring of a graph G we mean any proper vertex coloring consisting of the color classes C_1, …, C_k 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 {u_1, …, u_k} of vertices of G such that u_j ∈ C_j for any j ∈{1, …, k} and u_k is adjacent to u_j for each 1 ≤ j ≤ k with j ≠k, and (iii) for each i and j with i ≠ j, the vertex u_j has a neighbor in C_i. Denote by z(G) the maximum number of colors used in any z-coloring of G. Denote the Grundy and b-chromatic number 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 {G_n}_n=3^∞ of graphs such that z(G_n)=n but Γ(G_n)= b(G_n)=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.

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