Two Paths Joining Given Vertices in 2k-Edge-Connected Graphs

Abstract

Let k≥2 be an integer and G = (V(G), E(G)) be a k-edge-connected graph. For X⊆V(G), e(X) denotes the number of edges between X and V(G) − X. Let {s _i , t _i }⊆X _i ⊆V(G) (i=1,2) and X ₁∩X ₂=∅. We here prove that if k is even and e(X _i )≤2k−1 (i=1,2), then there exist paths P ₁ and P ₂ such that P _i joins s _i and t _i , V(P _i )⊆X _i (i=1,2) and G − E(P ₁∪P ₂) is (k−2)-edge-connected (for odd k, if e(X ₁)≤2k−2 and e(X ₂)≤2k−1, then the same result holds [10]), and we give a generalization of this result and some other results about paths not containing given edges.