Oriented diameter and rainbow connection number of a graph

Abstract

The oriented diameter of a bridgeless graph G is minfdiam(H) j H is a strang orientation of Gg. A path in an edge-colored graph G, where adjacent edges may have the same color, is called rainbow if no two edges of the path are colored the same. The rainbow connection number rc(G) of G is the smallest integer number k for which there exists a k-edge-coloring of G such that every two distinct vertices of G are connected by a rainbow path. In this paper, we obtain upper bounds for the oriented diameter and the rainbow connection number of a graph in terms of rad(G) and ν(G), where rad(G) is the radius of G and ν(G) is the smallest integer number such that every edge of G is contained in a cycle of length at most ν(G). We also obtain constant bounds of the oriented diameter and the rainbow connection number for a (bipartite) graph G in terms of the minimum degree of G. © 2014 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France.