On the Locating Rainbow Connection Number of A Graph

Abstract

Let k be a positive integer and G = (V(G), E(G)) be a finite and connected graph. A rainbow vertex k-coloring of G is a function c: V(G) ? {1,2,..., k} such that for every two vertices u and v in V(G) there exists a u-v path whose internal vertices have distinct colors. Such path is called a rainbow vertex path. The rainbow vertex connection number of G, denoted by rvc(G). is the smallest positive integer k so that G has a rainbow vertex k-coloring. The distance between two difference vertices u and v in V(G), denoted by d(u,v), is the length of a shortest u-v path in G. For i ? {1,2,..., k], let Ri be the set of vertices with color i and ? - {R 1,R 2,...,Rk } be fjn ordered partition of V(G). The rainbow code of a vertex v of V(G) with respect to ? is defined as the k-tuple rcn(v)=(d(v,{R}_{1}),d(v,{R}_{2}),d(v,{R}_{k})), where d(v, Ri ) =min{d(v, y) | y ? Ri } for each i ? {1,2,..., k}. If every vertex of G has distinct rainbow codes, then c is called a locating rainbow k-coloring of G. The locating rainbow connection number of G. denoted by rvcl(G). is defined as the smallest positive integer k such that G has a locating rainbow k-coloring. In this paper, we provide the sharp upper and lower bounds for locating rainbow connection number of a graph. We also determine the locating rainbow connection number of some well-known classes of graphs.