On the mixed connectivity conjecture of Beineke and Harary

The conjecture of Beineke and Harary states that for any two vertices which can be separated by k vertices and l edges for l≥ 1 but neither by k vertices and l-1 edges nor k-1 vertices and l edges there are k+l edge-disjoint paths connecting these two vertices of which k+1 are internally disjoint.In this paper we prove this conjecture for l=2 and every k∈ℕ .We utilize this result to prove that the conjecture holds for all graphs of treewidth at most 3 and all k and l .