The rainbow vertex connection number of edge corona product graphs

Abstract

Let G1, G2 be a special graphs with vertices of G1 1,2,…, n and edges of G1 1,2,… m. The generalized edge corona product of graphs G1 and G2, denoted by G1 ⋄ G1 is obtained by taking one copy of graph G1 and m copy of G2, thus for each edge ek = ij of G, joining edge between the two end-vertices i, j of ek and each vertex of the k-copy of G2. A rainbow vertex-coloring graph G where adjacent vertices u-v and its internal vertices have distinct colors. A path is called a rainbow path if no two verticess of the path have the same color. A rainbow vertex-connection number of graph G is minimun number of colors in graph G to connected every two distinct internal vertices u and v such that a graph G naturally rainbow vertex-connected, denoted by rvc (G). In this paper, we determine minimum integer for rainbow vertex coloring of edge corona product on cycle and path such as Pn ⋄ Pm, Pn ⋄ Cm, Cn ⋄ Pm, and Cn ⋄ Cm.