Graphs with vertex rainbow connection number two

• Published in: Science China. Mathematics Vol. 58; no. 8; pp. 1803 - 1810
• Main Authors: Lu, ZaiPing, Ma, YingBin
• Format: Journal Article
• Language: English
• Published: Beijing Science China Press 01-08-2015
• Subjects: Applications of Mathematics; Mathematics; Mathematics and Statistics; 彩虹; 最小数; 点连接; 着色图; 连接数; 钢筋混凝土; 顶点; 颜色; vertex rainbow connection number; clique number; 05C35; vertex-coloring; 05C40; 05C15
• ISSN: 1674-7283; 1869-1862
• DOI: 10.1007/s11425-014-4905-0

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 ｜E（G）｜ ≥ （n2-2） ＋ 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） = （n2-2） ＋ 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.