The lower bound of the r-dynamic chromatic number of corona product by wheel graphs

Abstract

The dynamic coloring of a graph G is proper coloring such that every vertex of G with degree has at least two neighbors that are colored differently. A generalization of the dynamic coloring was also introduced by Montgomery in [12], the generalized concept is called r-dynamic k-coloring. An r-dynamic coloring of a graph G is a proper coloring c of the vertices such that |c(N(v)| ≥ min{r, d(v)}, for each v ∈V(G). The r-dynamic chromatic number of a graph G, denoted χr(G) is the smallest k such that c is an r-dynamic k coloring of G. We will find the lower bound of the r-dynamic chromatic number of graphs corona wheel graph and some new results the exact value of r-dynamic chromatic number of corona graphs. In this paper, we study the lower bound of Xr(H⊙Wm), Xr(Wn⊙H) and we also prove the exact value of r-dynamic chromatic number of some graphs.