Local antimagic vertex dynamic coloring of some graphs family

All graphs in this paper are simple and connected graph. A vertex dynamic coloring is a proper vertex k-coloring of graph G such that |c(N(v_i))| ≥ min{r, d(v)} and the neighbourhood of vertex u has different colors. A bijection f : E(G) → {1, 2, 3, ..., m} is called a local antimagic dynamic coloring, such that: (1) if uv E(G), where w(u) = ∑_eE(u) f(e) and (2) for each vertex v V(G), |w(N(v_i))| ≥ min{r, d(v_i)}. The local antimagic vertex dynamic chromatic number denoted by ${\chi }_{r}^{la}(G)$ is the minimum number of colors needed to color G in such a way the graph G to be local antimagic vertex dynamic graph. In this paper, we will study the existence of the local antimagic vertex dynamic chromatic number of some graph classes, namely caterpilar, doublebroom, broom and sun graph.