On the Rainbow Vertex Connection Number of Edge Comb of Some Graph

Agustina M. (CGANT University of Jember Indonesia Mathematics Depart. University of Jember Indonesia)
Dafik Dafik (CGANT University of Jember Indonesia Mathematics Edu. Depart. University of Jember Indonesia)
Slamin Slamin (CGANT University of Jember Indonesia System Information. Depart. University of Jember)
Kusbudiono Kusbudiono (CGANT University of Jember Indonesia Mathematics Depart. University of Jember Indonesia)

UNEJ e-Proceeding
2016: Proceeding The 1st International Basic Science Conference

By an edge comb, we mean a graph formed by combining two graphs G and H, where each edge of graph G is replaced by the which one edge of graph H, denote by G D H. A vertex colored graph G D H = (V (G D H);E(G D H)) is said rainbow vertex-connected, if for every two vertices u and v in V (G D H), there is a u ???? v path with all internal vertices have distinct color. The rainbow vertex connection number of G D H, denoted by rvc(G D H) is the smallest number of color needed in order to make G D H rainbow vertex-connected. This research aims to find an exact value of the rainbow vertex connection number of exponential graph, namely rvc(G D H) when G D H are Pn D Btm, Sn D Btm, Ln D Btm, Fm;n D Btp, rvc(Pn D Sm), rvc(Cn D Sm), and rvc(Wn D Sm) Wn D Btm. The result shows that the resulting rainbow vertex connection attain the given lower bound.