Upper Bounds for the Rainbow Connection Numbers of Line Graphs

摘要

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 rc(G), is defined as the smallest number of colors such that G is rainbow connected. In this paper, we mainly study the rainbow connection number rc(L(G)) of the line graph L(G) of a graph G which contains triangles. We get two sharp upper bounds for rc(L(G)), in terms of the number of edge-disjoint triangles of G. We also give results on the iterated line graphs.