Graphs with vertex rainbow connection number two

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 graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. A vertex colored graph G is vertex rainbow connected if any two vertices are connected by a path whose internal vertices have distinct colors. The vertex rainbow connection number of G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G vertex rainbow connected. In 2011, Kemnitz and Schiermeyer considered graphs with rc(G) = 2. We investigate graphs with rvc(G) = 2. First, we prove that rvc(G) ⩽ 2 if \(\left| {E(G)} \right| \geqslant \left( {\begin{array}{*{20}c} {n - 2} \\ 2 \\ \end{array} } \right) + 2\), and the bound is sharp. Denote by s(n, 2) the minimum number such that, for each graph G of order n, we have rvc(G) ⩽ 2 provided |E(G)| ⩾ s(n, 2). It is proved that \(s(n,2) = \left( {\begin{array}{*{20}c} {n - 2} \\ 2 \\ \end{array} } \right) + 2\). Next, we characterize the vertex rainbow connection numbers of graphs G with |V (G)| = n, diam(G) ⩾ 3 and clique number ω(G) = n−s for 1 ⩽ s ⩽ 4.