On local vertex irregular reflexive coloring of graphs

Let χ(G) be a chromatic number of proper coloring on G. For an injection f : V (G) → {0, 2, . . . ; 2k_υg} and f : E(G) → }1, 2, . . . , k_e}, where k = max{k_e, 2k_χ} for k_υ, k_e are natural number. The associated weight of a vertex u, υ ∈ V (G) under f is w(u) = f(u) + ∑_uυ∈E(G)f(uυ). The function f is called a local vertex irregular reflexive k-labeling if every two adjacent vertices has distinct weight. When we assign each vertex of G with a color of the vertex weight w(uυ), thus we say the graph G admits a local vertex irregular reflexive coloring. The smallest number of vertex weights needed to color the vertices of G such that no two adjacent vertices share the same color is called a local vertex irregular reflexive chromatic number, denoted by _χirvs(G). Furthermore, the minimum k required such that _χlrvs(G) = χ(G) is called a local reflexive vertex color strength, denoted by lrvcs(G). In this paper, we will obtain the lrvcs(G) and characterize the existence of a graph with given its local reflexive vertex color strength.