On the equitable k *-laceability of hypercubes

Abstract

Let G be a finite undirected bipartite graph. Let u, v be two vertices of G from different partite sets. A collection of k internal vertex disjoint paths joining u to v is referred as a k-container Ck(u,v). A k-container is a k*-container if it spans all vertices of G. We define G to be a k*-laceable graph if there is a k*-container joining any two vertices from different partite sets. A k*-container Ck*(u,v)={P1,…,Pk} is equitable if ||V(Pi)|−|V(Pj)||≤2 for all 1≤i,j≤k. A graph is equitably k*-laceable if there is an equitable k*-container joining any two vertices in different partite sets. Let Qn be the n-dimensional hypercube. In this paper, we prove that the hypercube Qn is equitably k*-laceable for all k≤n−4 and n≥5.