3 - Chromatic Cubic Graphs with Complementary Connected Domination Number Three

Let G (V, E) be a graph. A subset S of V is called a dominating set of G if every vertex in V-S is adjacent to at least one vertex in S. The domination number γ (G) is the minimum cardinality taken over all such dominating sets in G. A subset S of V is said to be a complementary connected dominating set (ccd-set) if S is a dominating set and < V-S > is connected. The chromatic number χ is the minimum number of colours required to colour all the vertices such that no two adjacent vertices receive the same colour. In this paper, we characterize the r regular graphs with γcc = χ = 2 and the 3 regular graphs with γcc = χ = 3.