Upper bounds for the rainbow connection numbers of line graphs

TL;DR

This paper establishes sharp upper bounds for the rainbow connection number of line graphs, especially those containing triangles, and explores related properties of iterated line graphs.

Contribution

It provides new upper bounds for the rainbow connection number of line graphs based on edge-disjoint triangles in the original graph.

Findings

Two sharp upper bounds for $rc(L(G))$ in terms of edge-disjoint triangles.

Results on the rainbow connection number of iterated line graphs.

Abstract

A path in an edge-colored graph $G$, where adjacent edges may be colored the same, is called a rainbow path if no two edges of it are colored the same. A nontrivial connected graph $G$ is rainbow connected if for any two vertices of $G$ there is a rainbow path connecting them. The rainbow connection number of $G$, denoted by $rc(G)$, is defined as the smallest number of colors by using which there is a coloring such that $G$ is rainbow connected. In this paper, we mainly study the rainbow connection number of the line graph of a graph which contains triangles and get two sharp upper bounds for $rc(L(G))$, in terms of the number of edge-disjoint triangles of $G$ where $L(G)$ is the line graph of $G$. We also give results on the iterated line graphs.