On the total rainbow connection of the wheel related graphs
M. S. Hasan, Slamin Slamin, Dafik Dafik, Ika Hesti Agustin, Ridho Alfarisi · Journal of Physics Conference Series · 2018
Let G = ( V ( G ), E ( G )) be a nontrivial connected graph with an edge coloring c : E ( G ) → {1, 2, ..., l }, l ∊ N , with the condition that the adjacent edges may be colored by the same colors. A path P in G is called rainbow path if no two edges of P are colored the same. The smallest number of colors that are needed to make G rainbow edge-connected is called the rainbow edge-connection of G , denoted by rc ( G ). A vertex-colored graph is rainbow vertex-connected if any two vertices are connected by a path whose internal vertices have distinct colors. The smallest number of colors that are needed to make G rainbow vertex-connected is called the rainbow vertex-connection of G , denoted by rvc ( G ). A total-colored path is total-rainbow if edges and internal vertices have distinct colours. The minimum number of colour required to color the edges and vertices of G is called the total rainbow connection number of G , denoted by trc ( G ). In this paper, we determine the total rainbow connection number of some wheel related graphs such as gear graph, antiweb-gear graph, infinite class of convex polytopes, sunflower graph, and closed-sunflower graph.