THE RAINBOW VERTEX-CONNECTION NUMBERS OF WHEEL-SHIELD GRAPHS

Let be a nontrivial simple connected graph, be an edge of and be an integer greater than or equal to . A path of order , denoted by , is a graph whose vertices can be labelled such that . A -shield graph is a graph obtained by and copies of such that the edge of -th embedded to -th edge of by embedding to and to . A path in a vertex-colored graph is said to be rainbow-vertex path if every internal vertex in the path has different color. A vertex-colored graph is said to be rainbow-vertex connected if for every pair of vertices there exists a rainbow-vertex path connecting them. The rainbow- vertex connection number of , denoted by , is the minimum colors needed to make rainbow-vertex connected. In this paper, we determine the rainbow-vertex connection numbers of of wheel-shield graphs , specifically finding that the number ranges from to depending on the order of the wheel.