On r-dynamic local irregularity vertex coloring of special graphs

Abstract

In this paper, we study a new notion of graph coloring, that is a local irregularity vertex r-Dynamic coloring. We combine irregular local and the principles of r-dynamic coloring and we assign color to all vertices by using the weights of local irregularity vertex. We define l : V(G) → {1, 2, …, k} as a vertex irregular k-labeling and w : V(G) → N where w(u) = ∑ v∈N(u) l(v). By a local irregularity vertex coloring, we define a condition for f if for every uv ∈ E(G), w(u) ≠ w(v) and max(l) = min{max{li }; li vertex irregular labeling}. Each vertex of the weight as a color should satisfy the r-dynamic condition, namely |w(N(v)) ≥ min{r, d(v)} and each the adjacent vertices must be different. In this paper, we study the local irregularity vertex r-Dynamic coloring of special graph, namely triangular book graph, central of friendship graph, tapol graph, and rectangular book graph.