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In 2008, Chartrand et al. first introduced the concept of rainbow connection. Since then the study of rainbow connection has received considerable attention in the literature, and now it becomes an active topic in graph theory. As a natural generalization, Uchizawa et al. (2013) and Liu et al. (2014) presented the concept of total rainbow connection, respectively. In this paper, we investigate the total rainbow connection numbers of outerplanar graphs with diameter 2. Applying our result, we improve the main result of [X. Huang, X. Li, Y. Shi, J. Yue, Y. Zhao, Rainbow connections for outerplanar graphs with diameter 2 and 3, Applied Mathematics and Computation, 242 (2014), 277–280]. Next, we revise the main result of [Y. Liu, Z. Wang, Rainbow Connection Number of the Thorn Graph, Applied Mathematical Sciences, 8 (2014), 6373–6377], and determine the total rainbow connection numbers of graphs G , where G are the thorn graph of complete graph K n * , the thorn graph of the cycle C n *. At last, we study the rainbow 2-connection numbers of some special graphs. [ABSTRACT FROM AUTHOR]

A total-colored graph G is total rainbow connected if any two vertices are connected by a path whose edges and inner vertices have distinct colors. A graph G is total rainbow k -connected if there is a total-coloring of G with k colors such that G is total rainbow connected. The total rainbow connection number, denoted by trc ( G ), of a graph G is the smallest k to make G total rainbow k -connected. For n, k ≥ 1, define h ( n, k ) to be the minimum size of a total rainbow k -connected graph G of order n . In this paper, we prove a sharp upper bound for trc ( G ) in terms of the number of vertex-disjoint cycles of G . We also compute exact values and upper bounds for h ( n, k ). [ABSTRACT FROM AUTHOR]