On the number of edges in a graph with no (k+1)-connected subgraphs

Mader proved that for k >= 1 and n >= 2k, every n-vertex graph with no (k + 1)-connected subgraphs has at most (1 + 1/root 2) (n - k) edges. He also conjectured that for n large with respect to k, every such graph has at most 3/2 (k - 1/3) (n-k) edges. Yuster improved Mader's upper bound to 193/120k(n-k) for n >= 9k/4. In this note, we make the next step towards Mader's Conjecture: we improve Yuster's bound to 19/12k(n-k) for n >= 5k/2. (C) 2015 Elsevier B.V. All rights reserved.