On r-dynamic chromatic number of coronation of order two of any graphs with path graph

Abstract

In this study we assume every graphs be a simple, finite, nontrivial, connected and undirected graphs with vertex set V(H), edge set E(H) and no isolated vertex. An r-dynamic coloring of a graph H is a proper k-coloring of graph H such that the neighbors of any vertex u accept at least min{r, d(u)} different colors. The r-dynamic chromatic number is the minimum k such that graph H has an r-dynamic k-coloring, it is shown by χr(H). In this research, we interpret the r-dynamic chromatic number of graph corona of order two, denoted by H1⊙2 H2. The corona of two graphs H1 and H2 is a graph H1 ⊙ H2 formed from one copy of H1 and |V(H1)| copies of H2 where the ith vertex of H1 is adjacent to every vertex in the ith copy of H2. For any integer l ≥ 2, we establish the graph H1 ⊙l H2 recursively from H1 ⊙ H2 as H1 ⊙l H2 = (H1 ⊙l−1 H2) ⊙ H2. Graph H1 ⊙l H2 is also named as l − corona product of H1 and H2. Xr = (Pn ⊙2 H2) and H2 is path graph Pm, star graph Sm, complete graph Km and fan graph Fm. In this study we will determine the lower bound of r-dynamic chromatic number of corona order two of graphs.