Some upper bounds for the total-rainbow connection number of graphs

A path P in a total-colored graph G is total-rainbow if its edges and internal vertices have distinct colors, and G is total-rainbow connected if it has a total-rainbow path between every two vertices of G. The total-rainbow connection number of G, denoted by trc(G), is the minimum number of colors required to make G total-rainbow connected. In [12], the authors derived a tight upper bound for the total-rainbow connection number of 2-connected graphs. In this paper, we study the total-rainbow connection number of 1-connected graphs and obtain a tight upper bound for the total-rainbow connection number of a 1-connected graph in terms of its order and number of cut vertices. Besides, we investigate the total-rainbow connection number of the complement of a graph according to some constraints of the graph itself and obtain a tight Nordhaus-Gaddum-type upper bound for the total-rainbow connection number.