Coloracão arco-íris em classes de grafos - Rocha, Aleffer (2020)
Rainbow coloring problems, of noteworthy applications in Information Security, have been receiving much attention in the last years in Combinatorics. In particular, the rainbow connection number of a connected graph , denoted (), is the least for which admits a (not necessarily proper) -edge-coloring such that between any pair of vertices there is a rainbow path, i. e., a path whose edge colors are all distinct. Given an edge coloring for a graph , if between any pair of vertices there is a minimum path which is rainbow, then is a strong rainbow coloring. The minimum number of colors for which a given graph has a strong rainbow coloring is the strong rainbow connection number of . A graph is rainbow critical if the deletion of any edge of increases (). In this work, we determine the rainbow connection number and the strong rainbow connection number of shadow graphs of paths, triple triangular snake graphs, circulant graphs 1, 2 and join graphs + when has () ≤ 2. We also determine the rainbow connection number of join graphs of sunlet () with when | ()| ≥ − 1 and of cographs with three pendent vertices. We found new upper bounds for the rainbow connection number of snake graphs, join graphs of two graphs with universal vertices, and join graphs + when ≤ − 2. We also present necessary and sufficient conditions for the criticality of fan graphs, Cartesian products × when () = () ≥ 2 or when is a pan graph, × when () = () ≥ 1, and × when and are even.