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 r(G) is the smallest k such that c is an r-dynamic k coloring of G. We will find the lower bound of the rdynamic 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 𝜒𝑟 (𝐻⨀𝑊𝑚 ), 𝜒𝑟(𝑊𝑛⨀𝐻) and we also prove the exact value of r-dynamic chromatic number of some graphs.