(1, 2)-rainbow connection number at most 3 in connected dense graphs

Abstract

Let G be an edge-colored connected graph G . A path P in the graph G is called l -rainbow path if each subpath of length at most l + 1 is rainbow. The graph G is called ( k, l ) -rainbow connected if any two vertices in G are connected by at least k pairwise internally vertex-disjoint l -rainbow paths. The smallest number of colors needed in order to make G ( k, l ) -rainbow connected is called the ( k, l ) -rainbow connection number of G and denoted by rc k,l ( G ) . In this paper, we consider the (1 , 2) -rainbow connection number at most 3 in some connected dense graphs. Our main results are as follows: (1) Let n ≥ 7 be an integer and G be a connected graph of order n . If ω ( G ) ≥ n − 3 , then rc 1 , 2 ( G ) ≤ 3 . Moreover, the bound of the clique number is sharpness. (2) Let n ≥ 7 be an integer and G be a connected graph of order n . If | E ( G ) | ≥ (cid:0) n − 3 2 (cid:1) + 7 , then rc 1 , 2 ( G ) ≤ 3 .