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

Abstract

A path in an edge-colored graph is called {\em rainbow} if no two edges of it are colored the same. For an $\ell$-connected graph $G$ and an integer $k$ with $1\le k\le \ell$, the {\em 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 that what is the minimum constant $\alpha >0$ such that for all 2-connected graphs $G$ on $n$ vertices, we have $r{c}_2(G)\le \alpha n$. In this paper, we prove that $\alpha =1$ and $r{c}_2(G)=n$ if and only if $G$ is a cycle of order $n$, settling down this problem.

8 pages

References in corpus (1)

• Rainbow Connection Number and Radius

Cited by in corpus (1)

• Rainbow connections for planar graphs and line graphs