The total-rainbow k-connection of 2-connected graphs

A graph is said to be total-colored if all the edges and the vertices of the graph are colored. Let k be a positive integer. A path P in a total-colored graph G is called a total-rainbow path if its edges and internal vertices have distinct colors. The total-colored graph G is total-rainbow k-connected if any two vertices of G are connected by k disjoint total-rainbow paths. The total-rainbow k-connection number of G, denoted by trc(k)(G), is the minimum number of colors needed to make G total-rainbow k-connected. In this paper, we give tight upper bounds for the total-rainbow k-connection number trc(k)(G) of a 2-connected graph G. Moreover, trc(2)(G) = 2n (n >= 5) if and only if G is a cycle of order n.