On various (strong) rainbow connection numbers of graphs

An edge-coloured path is rainbow if all of its edges have distinct colours. For a connected graph G, the rainbow connection number rc(G) of G is the minimum number of colours in an edge-colouring of G such that, any two vertices are connected by a rainbow path. Similarly, the strong rainbow connection number src(G) ofG is the minimum number of colours in an edge-colouring of G such that, any two vertices are connected by a rainbow geodesic (i.e., a path of shortest length). These two concepts of connectivity in graphs were introduced by Chartrand et al. in 2008. Subsequently, vertex-coloured versions of both parameters, rvc(G) and srvc(G), and a total-coloured version of the rainbow connection number, trc(G), were introduced. In this paper we introduce the strong total ∗ Also at School of Mathematics and Statistics, Qinghai Normal University, Xining, Qinghai 810008, China † Corresponding author LIN CHEN ET AL. /AUSTRALAS. J. COMBIN. 70 (1) (2018), 137–156 138 rainbow connection number strc(G), which is the version of the strong rainbow connection number using total-colourings. Among our results, we will determine the strong total rainbow connection numbers of some special graphs. We will also compare the six parameters, by considering how close and how far apart they can be from one another. In particular, we will characterise all pairs of positive integers a and b such that, there exists a graph G with trc(G) = a and strc(G) = b, and similarly for the parameters rvc and srvc.