A note on graphs without k-connected subgraphs

Given integers k ≥ 2 and n ≥ k, let c(n, k) denote the maximum possible number of edges in an n-vertex graph which has no k-connected subgraph. It is immediate that c(n, 2) = n − 1. Mader [2] conjectured that for every k ≥ 2, if n is sufficiently large then c(n, k) ≤ (1.5k − 2)(n− k + 1), where equality holds whenever k − 1 divides n. In this note we prove that when n is sufficiently large then c(n, k) ≤ 193 120 (k − 1)(n − k + 1) < 1.61(k − 1)(n − k + 1), thereby coming rather close to the conjectured bound.