On the extremal sizes of maximal graphs without [formula omitted]-connected subgraphs

Abstract

Let G be a graph and let κ(G) be the vertex-connectivity of G. The maximum subgraph connectivity of G is κ¯(G)=max{κ(H):H⊆G}. A simple graph G is vertex-k-maximal if κ¯(G)≤k, but for any e∈E(Gc), κ¯(G+e)≥k+1. Mader conjectured that every vertex-k-maximal simple graph of order n satisfies |E(G)|≤32(k−13)(n−k).We prove the following. (i) Every vertex-k-maximal simple graph of order n satisfies |E(G)|≥(n−k)k+k(k−1)2. This lower bound is best possible. (ii) For every integer m in the range 2n−3≤m≤5n∕2−21+(−1)n+14 there exists a vertex 2-maximal graph of order n with m edges.