Grundy coloring in some subclasses of bipartite graphs and their complements

A vertex v is a Grundy vertex with respect to a proper k-coloring c of a graph G = (V, E) if v has a neighbor of color j for every j (1 <= j < i <= k), where i = c(v). A proper k -coloring c of G is called a Grundy k-coloring of G if every vertex is a Grundy vertex with respect to c and the largest integer k such that G admits a Grundy k-coloring is called the Grundy number of G which is denoted as Gamma(G). Given a graph G and an integer k, the Grundy number decision problem is to decide whether Gamma(G) >= k. The Grundy number decision problem is known to be NP-complete for bipartite graphs and complement of bipartite graphs. In this paper, we strengthen this result by showing that this problem remains NP-complete for perfect elimination bipartite graphs as well as for complement of perfect elimination bipartite graphs. Further, we give a linear-time algorithm to find the Grundy number of chain graphs, which is a proper subclass of the class of perfect elimination bipartite graphs. We also give a linear-time algorithm to find the Grundy number in complements of chain graphs. A partial Grundy coloring of a graph G is a proper k-coloring of G such that there is at least one Grundy vertex with each color i <= 1 < i <= k and the partial Grundy number of G, partial derivative Gamma yy(G), is the largest integer k such that G admits a partial Grundy k-coloring. Given a graph G and an integer k, the partial Grundy number decision problem is to decide whether partial derivative Gamma(G)>= k. It is known that the partial Grundy number decision problem is NP-complete for bipartite graphs. In this paper, we prove that this problem is NP-complete in the complements of bipartite graphs by showing that the Grundy number and partial Grundy number are equal in complements of bipartite graphs. (C) 2020 Elsevier B.V. All rights reserved.