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 ℓℓ-connected graph GG and an integer kk with 1≤k≤ℓ1≤k≤ℓ, the rainbow  kk-connection number  rck(G)rck(G) of GG is defined to be the minimum number of colors required to color the edges of GG such that every two distinct vertices of GG are connected by at least kk internally disjoint rainbow paths. Fujita et al. proposed a problem: What is the minimum constant α>0α>0 such that for every 2-connected graph GG on nn vertices, we have rc2(G)≤αnrc2(G)≤αn ? In this paper, we prove that the minimum constant α=1α=1 and rc2(G)=nrc2(G)=n if and only if GG is a cycle of order nn, which solves the problem of Fujita et al.