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

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 = (V1, . . . , Vk) such that each of Vi in an independent set. The chromatic number is the smallest k among all colorings C = (V1, . . . , Vk) of G and is denoted by χ(G). A vertex u ∈ V (G) is said to dominate the color class Vi if u is the unique vertex of Vi or if it is adjacent to all the vertices of Vi. A coloring C is said to be the dominator coloring if every vertex dominates some color class in C. The dominator chromatic number of G is the minimum k of all dominator colorings of G and is denoted by χd(G). Further, a graph G is D(k) if γ(G) = χ(G) = χd(G) = k. In 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 K2,q where q ≥ 2.