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Mutually independent bipanconnected property of hypercube

Authors
Journal
Applied Mathematics and Computation
0096-3003
Publisher
Elsevier
Publication Date
Volume
217
Issue
8
Identifiers
DOI: 10.1016/j.amc.2010.10.008
Keywords
  • Hypercubes
  • Panconnected
  • Mutually Independent

Abstract

Abstract A graph is denoted by G with the vertex set V( G) and the edge set E( G). A path P = 〈 v 0, v 1, … , v m 〉 is a sequence of adjacent vertices. Two paths with equal length P 1 = 〈 u 1, u 2, … , u m 〉 and P 2 = 〈 v 1, v 2, … , v m 〉 from a to b are independent if u 1 = v 1 = 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 l be a positive integer length, d G ( u, v) ⩽ l ⩽ ∣ V( G) − 1∣ with ( l − 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 l 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 Q n ( u , v ) ⩾ n - 1 , is ( n − 1, l)-mutually independent bipanconnected for every l , d Q n ( u , v ) ⩽ l ⩽ | V ( Q n ) - 1 | with ( l - d Q n ( u , v ) ) being even. As for d Q n ( u , v ) ⩽ n - 2 , it is also ( n − 1, l)-mutually independent bipanconnected if l ⩾ d Q n ( u , v ) + 2 , and is only ( l, l)-mutually independent bipanconnected if l = d Q n ( u , v ) .

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