The size of graphs with restricted rainbow $2$-connection number

Abstract

Let k be a positive integer, and G be a k-connected graph. An edge-coloured path is rainbow if all of its edges have distinct colours. The rainbow k-connection number of G, denoted by rc k (G), is the minimum number of colours in an edge-colouring of G such that, any two vertices are connected by k internally vertex-disjoint rainbow paths. The function rc k (G) was introduced by Chartrand, Johns, McKeon and Zhang in 2009, and has since attracted significant interest. Let t k (n, r) denote the minimum number of edges in a k-connected graph G on n vertices with rc k (G) ≤ r. Let s k (n, r) denote the maximum number of edges in a k-connected graph G on n vertices with rc k (G) ≥ r. The functions t 1 (n, r) and s 1 (n, r) have previously been studied by various authors. In this paper, we study the functions t 2 (n, r) and s 2 (n, r). We determine bounds for t 2 (n, r) which imply that t 2 (n, 2) = (1 + o(1))n log 2 n, and t 2 (n, r) is linear in n for r ≥ 3. We also provide some remarks about the function s 2 (n, r).