Proper Rainbow Connection Number of Graphs

Abstract

A path in an edge-coloured graph is called a rainbow path if its edges receive pairwise distinct colours. An edge-coloured graph is said to be rainbow connected if any two distinct vertices of the graph are connected by a rainbow path. The minimum k for which there exists such an edge-colouring is the rainbow connection number rc(G) of G. Recently, Bau et al. [Rainbow connectivity in some Cayley graphs, Australas. J. Combin. 71 (2018) 381-393] introduced this concept with the additional requirement that the edge-colouring must be proper. The proper rainbow connection number of G, denoted by prc(G), is the minimum number of colours needed in order to make it properly rainbow connected. Obviously, prc(G) ≥ max{rc(G), χ′(G)}. In this paper we first prove an improved upper bound prc(G) ≤ n for every connected graph G of order n ≥ 3. Next we show that the difference prc(G)-max{rc(G), χ′(G)} can be arbitrarily large. Finally, we present several sufficient conditions for graph classes satisfying prc(G) = χ′(G).