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, . . . ; 2 k 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.