# Upper bounds for the total rainbow connection of graphs

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## Abstract

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 \(\delta \ge \sqrt{n-2}-1\).

## Keywords

Total-colored graph Total rainbow connection Minimum degree 2-Step dominating set## Mathematics Subject Classification

05C15 05C40 05C69 05D40## Notes

### Acknowledgments

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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