Upper bounds for the total rainbow connection of graphs

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 $$G$$G, the total rainbow connection number of $$G$$G, denoted by $$trc(G)$$trc(G), is defined as the smallest number of colors that are needed to make $$G$$G total-rainbow connected. These concepts were introduced by Liu et al. Notice that for a connected graph $$G$$G, $$2diam(G)-1\le trc(G)\le 2n-3$$2diam(G)-1≤trc(G)≤2n-3, where $$diam(G)$$diam(G) denotes the diameter of $$G$$G and $$n$$n is the order of $$G$$G. In this paper we show, for a connected graph $$G$$G of order $$n$$n with minimum degree $$\delta $$ź, that $$trc(G)\le 6n/{(\delta +1)}+28$$trc(G)≤6n/(ź+1)+28 for $$\delta \ge \sqrt{n-2}-1$$źźn-2-1 and $$n\ge 291$$nź291, while $$trc(G)\le 7n/{(\delta +1)}+32$$trc(G)≤7n/(ź+1)+32 for $$16\le \delta \le \sqrt{n-2}-2$$16≤ź≤n-2-2 and $$trc(G)\le 7n/{(\delta +1)}+4C(\delta )+12$$trc(G)≤7n/(ź+1)+4C(ź)+12 for $$6\le \delta \le 15$$6≤ź≤15, where $$C(\delta )=e^{\frac{3\log ({\delta }^3+2{\delta }^2+3)-3(\log 3-1)}{\delta -3}}-2$$C(ź)=e3log(ź3+2ź2+3)-3(log3-1)ź-3-2. Thus, when $$\delta $$ź is in linear with $$n$$n, the total rainbow number $$trc(G)$$trc(G) is a constant. We also show that $$trc(G)\le 7n/4-3$$trc(G)≤7n/4-3 for $$\delta =3$$ź=3, $$trc(G)\le 8n/5-13/5$$trc(G)≤8n/5-13/5 for $$\delta =4$$ź=4 and $$trc(G)\le 3n/2-3$$trc(G)≤3n/2-3 for $$\delta =5$$ź=5. Furthermore, an example from Caro et al. shows that our bound can be seen tight up to additive factors when $$\delta \ge \sqrt{n-2}-1$$źźn-2-1.