The 3-rainbow index and connected dominating sets

Abstract

A tree in an edge-colored graph is said to be rainbow if no two edges on the tree share the same color. An edge-coloring of $$G$$ is called a 3-rainbow coloring if for any three vertices in $$G$$ , there exists a rainbow tree connecting them. The 3-rainbow index $$rx_3(G)$$ of $$G$$ is defined as the minimum number of colors that are needed in a 3-rainbow coloring of $$G$$ . This concept, introduced by Chartrand et al., can be viewed as a generalization of the rainbow connection. In this paper, we study the 3-rainbow index by using connected 3-way dominating sets and 3-dominating sets. We show that for every connected graph $$G$$ on $$n$$ vertices with minimum degree at least $$\delta \, (3\le \delta \le 5)$$ , $$rx_{3}(G)\le \frac{3n}{\delta +1}+4$$ , and the bound is tight up to an additive constant; whereas for every connected graph $$G$$ on $$n$$ vertices with minimum degree at least $$\delta \, (\delta \ge 3)$$ , we get that $$rx_{3}(G)\le \frac{\ln (\delta +1)}{\delta +1}(1+o_{\delta }(1))n+5$$ . In addition, we obtain some tight upper bounds of the 3-rainbow index for some special graph classes, including threshold graphs, chain graphs and interval graphs.