The r-Dynamic Local Irregularity Vertex Coloring of Graph

Abstract

We define the r-dynamic local irregularity vertex coloring. Suppose  : V(G)  {1,2, … , k} is called vertex irregular k-labeling and w : V(G)  N where 𝑤(𝑢) = ∑ (𝑣) .  is called r-dynamic local irregular vertex coloring, if: (i) opt() = min{max{i}; i vertex irregular k-labeling}, (ii) for every ( ) uv  E(G), w(u) ≠ w(v), and (iii) for every v  V(G) such that |w(N(v))|  min{r, d(v)}. The chromatic number r-dynamic local irregular denoted by 𝜒

is minimum of cardinality r-dynamic local irregular vertex coloring. We study the r-dynamic local irregularity vertex coloring of graph and we have found the exact value of chromatic number r-dynamic local irregularity of some graph.