ON PROPER (STRONG) RAINBOW CONNECTION OF GRAPHS

A path in an edge-colored graph G is called a rainbow path if no two edges on the path have the same color. The graph G is called rainbow connected if between every pair of distinct vertices of G, there is a rainbow path. Recently, Johnson et al. considered this concept with the additional requirement that the coloring of G is proper. The proper rainbow connection number of G, denoted by pr c(C), is the minimum number of colors needed to properly color the edges of G so that G is rainbow connected. Similarly, the proper strong rainbow connection number of G, denoted by psr c(G), is the minimum number of colors needed to properly color the edges of G such that for any two distinct vertices of G, there is a rainbow geodesic (shortest path) connecting them. In this paper, we characterize those graphs with proper rainbow connection numbers equal to the size or within 1 of the size. Moreover, we completely solve a question proposed by Johnson et al. by proving that if G = K-p1 square ... square K-pn, where n >= 1, and p(1), ..., p(n)> 1 are integers, then pr c(G) = psr c(G) = chi'(G), where chi'(G) denotes the chromatic index of G. Finally, we investigate some sufficient conditions for a graph G to satisfy pr c(G) = rc(G), and make some slightly positive progress by using a relation between rc(G) and the girth of the graph.