Topological Properties of Hypercubes
IEEE Transactions on Computers
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CEA'09 Proceedings of the 3rd WSEAS international conference on Computer engineering and applications
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| Hi-index | 5.23 |
A bipartite graph G is bipanconnected if, for any two distinctvertices x and y of G, it contains an[x,y]-path of length l for each integer l satisfyingdG(x,y)≤/≤|V(G)|-1 and2|(l-dG(x,y)), wheredG(x,y) denotes the distance betweenvertices x and y in G and V(G) denotesthe vertex set of G. We say a bipartite graph G isbipanpositionably bipanconnected if, for any two distinct verticesx and y of G and for any vertexz∈V(G)-{x,y}, it contains a pathPl,k of length l such that x is thebeginning vertex of Pl,k, z is the(k+1)-th vertex of Pl,k, and y isthe ending vertex of Pl,k for each integerl satisfyingdG(x,z)+dG(y,z)≤/≤|V(G)|-1and 2|(l-dG(x,z)-dG(y,z)) and for eachinteger k satisfyingdG(x,z)≤k≤l-dG(y,z)and 2|(k-dG(x,z)). In this paper, we prove thatan n-cube is bipanpositionably bipanconnected ifn≥4. As a consequence, many properties of hypercubes,such as bipancyclicity, bipanconnectedness, bipanpositionableHamiltonicity, etc., follow directly from our result.