On Minimally (n, λ)-Connected Graphs

A graph G is (n, λ)-connected if it satisfies the following conditions: (1) |V(G)|⩾n+1; (2) for any subset S⊆V(G) and any subset L⊆E(G) with λ |S|+|L|<nλ, G−S−L is connected. The (n, λ)-connectivity is a common extension of both the vertex-connectivity and the edge-connectivity. An (n, 1)-connected graph is an n-(vertex)-connected graph, and a (1, λ)-connected graph is a λ-edge-connected graph. An (n, λ)-connected graph G is said to be minimally (n, λ)-connected if for any edge e in E(G), G−e is not (n, λ)-connected. Let G be a minimally (n, λ)-connected graph and let W be the set of its vertices of degree more than nλ. Then we first prove that for any subset W′ of W, the minimum degree of the subgraph of G induced by the vertex set W′ is less than or equal to λ. This result is an extension of a theorem of Mader, which states that the subgraph of a minimally n-connected graph induced by the vertices of degree more than n is a forest. By using our result, we show that if G is a minimally (n, λ)-connected graph, then (1) |E(G)|⩽λ(|V(G)|+n)2/8 for n+1⩽|V(G)|⩽3n−2; (2) |E(G)|⩽nλ(|V(G)|−n) for |V(G)|⩾3n−1. Furthermore, we study the number of vertices of degree nλ in a minimally nλ-connected graph.