# Frontiers of Mathematics in China

 Front. Math. China    2017, Vol. 12 Issue (4) : 921-936     DOI: 10.1007/s11464-017-0651-2
 RESEARCH ARTICLE |
Graphs with small total rainbow connection number
Yingbin MA1(), Lily CHEN2, Hengzhe LI1
1. College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China
2. School of Mathematics Science, Huaqiao University, Quanzhou 362021, China
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 Abstract A total-colored path is total rainbow if its edges and internal vertices have distinct colors. A total-colored graph G is total rainbow connected if any two distinct vertices are connected by some total rainbow path. The total rainbow connection number of G, denoted by trc(G), is the smallest number of colors required to color the edges and vertices of G in order to make G total rainbow connected. In this paper, we investigate graphs with small total rainbow connection number. First, for a connected graph G, we prove that $trc(G)=3?if(n−12)+1≤|E(G)|≤(n2)−1,$ and $trc(G)=6?if(n−22)+2≤.$ Next, we investigate the total rainbow connection numbers of graphs G with $|V(G)|=n,$ diam$(G)≥2,$ and clique number $ω(G)=n−s?for?1?≤s≤?3$. In this paper, we find Theorem 3 of [Discuss. Math. Graph Theory, 2011, 31(2): 313–320] is not completely correct, and we provide a complete result for this theorem. Corresponding Authors: Yingbin MA Issue Date: 06 July 2017
 Cite this article: Yingbin MA,Lily CHEN,Hengzhe LI. Graphs with small total rainbow connection number[J]. Front. Math. China, 2017, 12(4): 921-936. URL: http://journal.hep.com.cn/fmc/EN/10.1007/s11464-017-0651-2 http://journal.hep.com.cn/fmc/EN/Y2017/V12/I4/921