Rainbow Total-Coloring of Complementary Graphs and Erdős-Gallai Type Problem For The Rainbow Total-Connection Number

A total-colored graph $G$ is rainbow total-connected if any two vertices of $G$ are connected by a path whose edges and internal vertices have distinct colors. The rainbow total-connection number, denoted by $ rtc(G) $, of a graph $G$ is the minimum number of colors needed to make $G$ rainbow total-connected. In this paper, we prove that $ rtc(G) $ can be bounded by a constant 7 if the following three cases are excluded: $ diam( \overline{G} ) = 2 $, $ diam( \overline{G} ) = 3 $, $ \overline{G} $ contains exactly two connected components and one of them is a trivial graph. An example is given to show that this bound is best possible. We also study Erdős-Gallai type problem for the rainbow total-connection number, and compute the lower bounds and precise values for the function $ f(n, k) $, where $ f(n, k) $ is the minimum value satisfying the following property: if $ |E(G)| \ge f(n, k) $, then $ rtc(G) \le k $.