Total rainbow k-connection in graphs

Abstract Let k be a positive integer and G be a k -connected graph. In 2009, Chartrand, Johns, McKeon, and Zhang introduced the rainbow k -connection number r c k ( G ) of G . An edge-coloured path is rainbow if its edges have distinct colours. Then, r c k ( G ) is the minimum number of colours required to colour the edges of G so that any two vertices of G are connected by k internally vertex-disjoint rainbow paths. The function r c k ( G ) has since been studied by numerous researchers. An analogue of the function r c k ( G ) involving vertex colourings, the rainbow vertex k -connection number r v c k ( G ) , was subsequently introduced. In this paper, we introduce a version which involves total colourings. A total-coloured path is total-rainbow if its edges and internal vertices have distinct colours. The total rainbow k -connection number of G , denoted by t r c k ( G ) , is the minimum number of colours required to colour the edges and vertices of G , so that any two vertices of G are connected by k internally vertex-disjoint total-rainbow paths. We study the function t r c k ( G ) when G is a cycle, a wheel, and a complete multipartite graph. We also compare the functions r c k ( G ) , r v c k ( G ) , and t r c k ( G ) , by considering how close and how far apart t r c k ( G ) can be from r c k ( G ) and r v c k ( G ) .