The Mutually Independent Bipanconnected Property for Hypercube

A graph is denoted by G with the vertex set V(G) and the edge set E(G). A path P = (v₀, v₁, ¿, v_m) is a sequence of adjacent vertices. Two paths with equal length P₁ = (u₁, u₂, ¿, u_m) and P₂ = (v₁, v₂, ¿, v_m) from a to b are independent if u₁ = v₁ = a, u_m = v_m = b, and u_i ¿ v_i for 2 ¿ i ¿ m - 1. Paths with equal length {P_i}_i=1 ⁿ from a to b are mutually independent if they are pairwisely independent. Let u and v be two distinct vertices of a bipartite graph G, and let I be a positive integer length, d_G(u,v) ¿ I ¿ |V(G) - 1| with (I - d_G(u,v)) being even. We say that the pair of vertices u, v is (m, l)-mutually independent bipanconnected if there exist m mutually independent paths {P_i ^l}_i=1 ^m with length I from u to v. In this paper, we explore yet another strong property of the hypercubes. We prove that every pair of vertices u and v in the n-dimensional hypercube, with d_Qn (u, v) ¿ n - 1, is (n - 1, l)-mutually independent bipanconnected for every I, d_Qn (u, v) ¿ I ¿ |V(Q_n) - 1| with (I - d_Qn (u, v)) being even. As for d_Qn(u,v) ¿ n - 2, it is also (n - 1,l)-mutually independent bipanconnected if I ¿ d_Qn(u,v) + 2, and is only (Z,Z)-mutually independent bipanconnected if I = d_Qn(u,v).