Upper Bounds for Rainbow 2-Connectivity of the Cartesian Product of a Path and a Cycle

A path P in an edge-colored graph G where adjacent edges may be colored the same is said to be a rainbow path, if its edges have distinct colors. For a κ-connected graph G and an integer k with 1 ≤ k ≤ κ, the rainbow k-connectivity, rck (G) of G is defined as the minimum integer j for which there exists a j-edge-coloring of G such that every two distinct vertices of G are connected by k internally disjoint rainbow paths. In this paper, we determine upper bounds for rainbow 2-connectivity of the Cartesian product of two paths and the Cartesian product of a cycle and a path.