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.