Bipancyclic properties of faulty hypercubes
Summary: A bipartite graph \(G = (V, E)\) is bipancyclic if it contains cycles of every even length from 4 to \(|V|\) and edge bipancyclic if every edge lies on a cycle of every even length from 4 to \(|V|\). Let \(Q_n\) denote the \(n\)-dimensional hypercube. Let \(F\) be a subset of \(V(Q_n) \cup E(Q_n)\) such that \(F\) can be decomposed into two parts \(F_{av}\) and \(F_e\), where \(F_{av}\) is a union of \(f_{av}\) disjoint adjacent pairs of \(V(Q_n)\), and \(F_e\) consists of \(f_e\) edges. We prove that \(Q_n - F\) is bipancyclic if \(f_{av} + f_e \leq n - 2\). Moreover, \(Q_n - F\) is edge bipancyclic if \(f_{av} + f_e \leq n - 2\) with \(f_{av} < n - 2\).
scientific article; zbMATH DE number 6137695