ON DYNAMIC CHROMATIC NUMBER OF 4-REGULAR GRAPHS WITH GIRTH 3 AND 4

A proper vertex k -coloring of a graph G is called dynamic if for every vertex v ∈ V (G) with d (v) ≥ 2 the neighbors of v receive at least two different colors. The smallest integer k such that G has a k -dynamic coloring is called the dynamic chromatic number of G and denoted by xd (G) . In this paper, we show that the dynamic chromatic number of 4-regular graph with girth 3 and 4 for n ≥ 6 . Also, we provide some relation between the domination number and chromatic number of 4-regular graph with girth 3 for n ≥ 6 Nordhaus - Gaddum type results are also obtained for these parameters.