2-Reducible Two Paths and an Edge Constructing a Path in (2k + 1)-Edge-Connected Graphs

Let k ≥ 5 be an odd integer and G = (V(G), E(G)) be a k-edge-connected graph. For ${Xsubseteq V(G),e(X)}$ denotes the number of edges between X and V(G) ? X. We here prove that if ${{s_i,t_i}subseteq X_isubseteq V(G)(i=1,2),f}$ is an edge between s ₁ and ${s_2,X_1cap X_2=emptyset,e(X_1)le 2k-3,e(X_2)le 2k-2}$ , and e(Y) ≥ k + 1 for each ${Ysubseteq V(G)}$ with ${Ycap{s_1,t_1,s_2,t_2}={s_1,t_2}}$ , 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 ${G-f-E(P_1cup P_2)}$ is (k ? 2)-edge-connected, and in fact we give a generalization of this result.