Rainbow Vertex Connection Numbers and Total Rainbow Connection Numbers of Middle and Total Graphs

Abstract

A vertex-colouring of a graph Γ is rainbow vertex connected if every pair of vertices ( u , v ) in Γ there is a u − v path whose internal vertices have different colours. The rainbow vertex connection number of a graph Γ , is the minimum number of colours needed to make Γ rainbow vertex connected, denoted by r v c ( Γ ) . Here, we study the rainbow vertex connection numbers of middle and total graphs. A total-colouring of a graph Γ is total rainbow connected if every pair of vertices ( u , v ) in Γ there is a u − v path whose edges and internal vertices have different colours. The total rainbow connection number of Γ , is the minimum number of colours required to colour the edges and vertices of Γ in order to make Γ total rainbow connected, denoted by t r c ( Γ ) . In this paper, we also research the total rainbow connection numbers of middle and total graphs.