Rainbow 2-connectivity of edge-comb product of a cycle and a Hamiltonian graph

An edge-colored graph G is rainbow k -connected, if for every two vertices of G , there are k internally disjoint rainbow paths, i.e., if no two edges of each path are colored the same. The minimum number of colors needed for which there exists a rainbow k -connected coloring of G , r c k ( G ) , is the rainbow k -connection number of G . Let G and H be two connected graphs, where O is an orientation of G . Let e → be an oriented edge of H . The edge-comb product of G (under the orientation O ) and H on e → , G o ⊳ e → H , is a graph obtained by taking one copy of G and | E ( G )| copies of H and identifying the i -th copy of H at the edge e → to the i -th edge of G , where the two edges have the same orientation. In this paper, we provide sharp lower and upper bounds for rainbow 2-connection numbers of edge-comb product of a cycle and a Hamiltonian graph. We also determine the rainbow 2-connection numbers of edge-comb product of a cycle with some graphs, i.e. complete graph, fan graph, cycle graph, and wheel graph.