Journal of Combinatorial Optimization

, Volume 32, Issue 1, pp 260–266

# Upper bounds for the total rainbow connection of graphs

• Hui Jiang
• Xueliang Li
• Yingying Zhang
Article

## 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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