On graphs whose domination number is equal to chromatic and dominator chromatic numbers

Abstract

For a graph G = (V (G), E(G)), a dominating set D is a vertex subset of V (G) in which every vertex of V (G) \ D is adjacent to a vertex in D. The domination number of G is the minimum cardinality of a dominating set of G and is denoted by γ(G). A coloring of G is a partition C = (V 1 , ..., V k ) such that each of V i in an independent set. The chromatic number is the smallest k among all colorings C = (V 1 , ..., V k ) of G and is denoted by χ(G). A coloring C = (V 1 , ..., V k ) is said to be dominator if, for all V i , every vertex v ∈ V i is singleton in V i or is adjacent to every vertex of V j . The dominator chromatic number of G is the minimum k of all dominator colorings of G and is denoted byIn this paper, for n ≥ 4k − 3, we prove that there always exists a D(k) graph of order n. We further prove that there is no planar D(k) graph when k ∈ {3, 4}. Namely, we prove that, for a non-trivial planar graph G, the graph G is D(k) if and only if G is K 2,q where q ≥ 2.