The Vertex-Rainbow Connection Number Of Some Graph Operations

A path in an edge-colored (respectively vertex-colored) graph G is rainbow (respectively vertex-rainbow) if no two edges (respectively internal vertices) of the path are colored the same. An edge-colored (respectively vertex-colored) graph G is rainbow connected (respectively vertex-rainbow connected) if every two distinct vertices are connected by a rainbow (respectively vertex-rainbow) path. The rainbow connection number rc(G) (respectively vertex-rainbow connection number rvc(G)) of G is the smallest number of colors that are needed in order to make G rainbow connected (respectively vertex-rainbow connected). In this paper, we show that for a connected graph G and any edge e = xy is an element of E(G), rvc(G) <= rvc(G - e) <= rvc(G) d(G-e)(x, y) - 1 if G - e is connected. For any two connected, nontrivial graphs G and H, rad(G square H) -1 <= rad(G square H) <2rad(GIIIH), where Gq H is the Cartesian product of G and H. For any two non-trivial graphs G and H such that G is connected, rvc(G o H) = 1 if diam(G o H) <= 2, rad(G) -1 <= rvc(G o H) <= 2rad(G) if diam(G) > 2, where G o H is the lexicographic product of G and H. For the line graph L(G) of a graph G we show that rvc(L(G)) <= rc(G), which is the first known nontrivial inequality between the rainbow connection number and vertex-rainbow connection number. Moreover, the bounds reported are tight or tight up to additive constants.