Rainbow Connection Number and the Number of Blocks

Abstract

An edge-colored graph G is rainbow connected if every pair of vertices of G are connected by a path whose edges have distinct colors. The rainbow connection number rc(G) of G is defined to be the minimum integer t such that there exists an edge-coloring of G with t colors that makes G rainbow connected. For a graph G without any cut vertex, i.e., a 2-connected graph, of order n, it was proved that $${rc(G)\leq\lceil\frac{n}{2} \rceil}$$ r c ( G ) ≤ ? n 2 ? and the bound is tight. In this paper, we prove that for a connected graph G of order n with at least one cut vertex, $${rc(G) \leq\frac{n+r-1} 2}$$ r c ( G ) ≤ n + r - 1 2 , where r is the number of blocks of G with even orders, and the upper bound is tight. Moreover, we also obtain a tight upper bound $${\lfloor(2n-2)/3\rfloor}$$ ? ( 2 n - 2 ) / 3 ? for the rainbow connection number of a bridgeless (2-edge-connected) graph of order n.