Rainbow Connection Numbers for Undirected Double-Loop Networks

Abstract

An edge-colored graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. In this paper, we investigate rainbow connection numbers of three subfamilies of undirected double-loop networks, denoted by rc(G(n; ±s ₁, ±s ₂)), where 1 ≤ s ₁ < s ₂ < n∕2. We almost determine the precise value for the case that $s_{1} = 1,s_{2} = 2$. For the case that $n = ks,s_{1} = 1,s_{2} = s$, where s ≥ 3 and k ≥ 1 are integers, we derive that $rc(G(ks;\pm 1,\pm s)) \leq \min \{\lceil \frac{k} {2} \rceil + s,\lceil \frac{s+1} {2} \rceil + k - 1\}$. For the case that $n = 2ks,s_{1} = 2,s_{2} = s$, where k ≥ 1 are integers and s ≥ 3 are odd integers, we have $rc(G(n;\pm s_{1},\pm s_{2})) \leq \lceil \frac{ks} {2} \rceil + k$.