Journal of Combinatorial Optimization

, Volume 32, Issue 1, pp 260–266 | Cite as

Upper bounds for the total rainbow connection of graphs



A total-colored graph is a graph such that both all edges and all vertices of the graph are colored. A path in a total-colored graph is a total rainbow path if its edges and internal vertices have distinct colors. A total-colored graph is total-rainbow connected if any two vertices of the graph are connected by a total-rainbow path of the graph. For a connected graph \(G\), the total rainbow connection number of \(G\), denoted by \(trc(G)\), is defined as the smallest number of colors that are needed to make \(G\) total-rainbow connected. These concepts were introduced by Liu et al. Notice that for a connected graph \(G\), \(2diam(G)-1\le trc(G)\le 2n-3\), where \(diam(G)\) denotes the diameter of \(G\) and \(n\) is the order of \(G\). In this paper we show, for a connected graph \(G\) of order \(n\) with minimum degree \(\delta \), that \(trc(G)\le 6n/{(\delta +1)}+28\) for \(\delta \ge \sqrt{n-2}-1\) and \(n\ge 291\), while \(trc(G)\le 7n/{(\delta +1)}+32\) for \(16\le \delta \le \sqrt{n-2}-2\) and \(trc(G)\le 7n/{(\delta +1)}+4C(\delta )+12\) for \(6\le \delta \le 15\), where \(C(\delta )=e^{\frac{3\log ({\delta }^3+2{\delta }^2+3)-3(\log 3-1)}{\delta -3}}-2\). Thus, when \(\delta \) is in linear with \(n\), the total rainbow number \(trc(G)\) is a constant. We also show that \(trc(G)\le 7n/4-3\) for \(\delta =3\), \(trc(G)\le 8n/5-13/5\) for \(\delta =4\) and \(trc(G)\le 3n/2-3\) for \(\delta =5\). Furthermore, an example from Caro et al. shows that our bound can be seen tight up to additive factors when \(\delta \ge \sqrt{n-2}-1\).


Total-colored graph Total rainbow connection Minimum degree 2-Step dominating set 

Mathematics Subject Classification

05C15 05C40 05C69 05D40 



The authors are very grateful to the referees for their helpful comments and suggestions. This study is supported by NSFC No.11371205, the “973” program No.2013CB834204, and PCSIRT


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Center for Combinatorics and LPMC-TJKLCNankai UniversityTianjinChina

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