A sharp upper bound for the rainbow 2-connection number of a 2-connected graph

A path in an edge-colored graph is called rainbow if no two edges of it are colored the same. For an ℓ -connected graph G and an integer k with 1 ≤ k ≤ ℓ , the rainbow k -connection number r c k ( G ) of G is defined to be the minimum number of colors required to color the edges of G such that every two distinct vertices of G are connected by at least k internally disjoint rainbow paths. Fujita et al. proposed a problem: What is the minimum constant α > 0 such that for every 2-connected graph G on n vertices, we have r c 2 ( G ) ≤ α n ? In this paper, we prove that the minimum constant α = 1 and r c 2 ( G ) = n if and only if G is a cycle of order n , which solves the problem of Fujita et al.