On extremal k-outconnected graphs

Let G be a minimally k-connected graph with n nodes and m edges. Mader proved that if n ≥ 3 k - 2 then m ≤ k (n - k), and for n ≥ 3 k - 1 an equality is possible if, and only if, G is the complete bipartite graph K_{k, n - k}. Cai proved that if n ≤ 3 k - 2 then m ≤ ⌊ (n + k)² / 8 ⌋, and listed the cases when this bound is tight. In this paper we prove a more general theorem, which implies similar results for minimally k-outconnected graphs; a graph is called k-outconnected from r if it contains k internally disjoint paths from r to every other node.