On the mixed connectivity conjecture of Beineke and Harary

Abstract

AbstractThe conjecture of Beineke and Harary states that for any two vertices which can be separated by k vertices and l edges for $$l\ge 1$$ l ≥ 1 but neither by k vertices and $$l-1$$ l - 1 edges nor $$k-1$$ k - 1 vertices and l edges there are $$k+l$$ k + l edge-disjoint paths connecting these two vertices of which $$k+1$$ k + 1 are internally disjoint.In this paper we prove this conjecture for $$l=2$$ l = 2 and every $$k\in \mathbb {N}$$ 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.