On CD-chromatic number and its lower bound in some classes of graphs

Abstract

A k-class domination coloring (k-cd-coloring) is a partition of the vertex set of a graph G into k independent sets V1,…,Vk, where each Vi is dominated by some vertex ui of G. The least integer k such that G admits a k-cd-coloring is called the cd-chromatic number, χcd(G), of G. A subset S of the vertex set of a graph G is called a subclique in G if dG(x,y)≠2 for every x,y∈S. The cardinality of a maximum subclique in G is called the subclique number, ωs(G), of G. In this paper, we present algorithms to compute an optimal cd-coloring and a maximum subclique of (i) trees with time complexity O(n) and (ii) co-bipartite graphs with time complexity O(n2.5). This improves O(n3) algorithms by Shalu et al. (2017, 2020). In addition, we prove tight upper bounds for the subclique number of the class of (i) P5-free graphs and (ii) double-split graphs.