More on foxes

An edge in a k-connected graph G is called k-contractible if the graph G/e obtained from G by contracting e is k-connected. Generalizing earlier results on 3-contractible edges in spanning trees of 3-connected graphs, we prove that (except for the graphs Kk+1 if k ∈ {1, 2}) (a) every spanning tree of a k-connected triangle free graph has two k-contractible edges, (b) every spanning tree of a k-connected graph of minimum degree at least 3 2 k− 1 has two k-contractible edges, (c) for k > 3, every DFS tree of a kconnected graph of minimum degree at least 3 2 k− 3 2 has two k-contractible edges, (d) every spanning tree of a cubic 3-connected graph nonisomorphic to K4 has at least 1 3 |V (G)| − 1 many 3-contractible edges, and (e) every DFS tree of a 3-connected graph nonisomorphic to K4, the prism, or the prism plus a single edge has two 3-contractible edges. We also discuss in which sense these theorems are best possible. AMS classification: 05c40, 05c05.