Partial Grundy Coloring in Some Subclasses of Bipartite Graphs and Chordal Graphs

Abstract

A proper k-coloring of \(G=(V,E)\) is an assignment of k colors to vertices of G such that no two adjacent vertices receive the same color. A proper k-coloring of a graph \(G = (V,E)\) partitions V into independent sets or color classes \(V_1,V_2,\ldots ,V_k\). A vertex \(v \in V_i\) is a Grundy vertex if it is adjacent to at least one vertex in each color class \(V_j\) for every \(j < i\). A coloring is a partial Grundy coloring if every color class has at least one Grundy vertex in it and the partial Grundy number, \(\delta \varGamma (G)\) of a graph G is the maximum number of colors used in a partial Grundy coloring. Given a graph G and an integer \(k (1 \le k \le n)\), the Partial Grundy Number Decision problem is to decide whether \(\delta \varGamma (G) \ge k\). It is known that the Partial Grundy Number Decision problem is NP-complete for bipartite graphs. In this paper, we strengthen this result by proving that this problem remains NP-complete even for perfect elimination bipartite graphs, a proper subclass of bipartite graphs. On the positive side, we propose a linear time algorithm to determine the partial Grundy number of a chain graph, a proper subclass of perfect elimination bipartite graphs. It is also known that the Partial Grundy Number Decision problem is NP-complete for (disconnected) chordal graphs. We strengthen this result by proving that the Partial Grundy Number Decision problem remains NP-complete even for (connected) doubly chordal graphs, a proper subclass of chordal graphs. On the positive side, we propose a linear time algorithm to determine the partial Grundy number of split graphs, a well known subclass of chordal graphs.