A New Property of Interconnection Networks

A graph G is pancyclic if G includes cycles of all lengths and G is edge-pancyclic if each edge lies on cycles of all lengths. A bipartite graph is edge-bipancyclic if each edge lies on cycles of every even length from 4 to |V (G)|. Two cycles with the same length m, C1 = ⟨u1, u2, · · · , um, u1⟩ and C2 = ⟨v1, v2, · · · , vm, v1⟩ passing through an edge (x, y) are independent with respect to the edge (x, y) if u1 = v1 = x, um = vm = y and ui ̸= vi for 2 ≤ i ≤ m − 1. Cycles with equal length C1, C2, · · · , Cn passing through an edge (x, y) are mutually independent with respect to the edge (x, y) if each pair of them are independent with respect to the edge (x, y). We propose a new concept called mutually independent edge-bipancyclicity. We say that a bipartite graph G is k-mutually independent edge-bipancyclic if for each edge (x, y) ∈ E(G) and for each even length l, 4 ≤ l ≤ |V (G)|, there are k cycles with the same length l passing through edge (x, y), and these k cycles are mutually independent with respect to the edge (x, y). In this paper, we prove that the hypercube Qn is (n − 1)-mutually independent edgebipancyclic for n ≥ 4.