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

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 χ_r(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. X_r = (P_n ⊙² H₂) and H₂ is path graph P_m, star graph S_m, complete graph K_m and fan graph F_m. In this study we will determine the lower bound of r-dynamic chromatic number of corona order two of graphs.