Two Paths Joining Given Vertices in \((2k + 1)\)-Edge-Connected Graphs

Abstract

Let \(k \geq 3\) be odd and \(G = (V(G), E(G))\) be a \(k\)-edge-connected graph. For \(X \subseteq V(G)\), \(e(X)\) denotes the number of edges between \(X\) and \(V(G) – X\). We here prove that if \(\{s_i, t_i\} \subseteq X_i \subseteq V(G)\), \(i = 1, 2\), \(X_1 \cap X_2 = \emptyset\), \(e(X_1) \leq 2k-2\) and \(e(X_2) < 2k-1\), then there exist paths \(P_1\) and \(P_2\) such that \(P_i\) joins \(s_i\) and \(t_i\), \(V(P_i) \subseteq X_i\) (\(i = 1, 2\)) and \(G – E(P_1 \cup P_2)\) is \((k-2)\)-edge-connected. And in fact, we give a generalization of this result and some other results about paths not containing given edges.