Hamiltonian numbers of Möbius double loop networks

  • Authors:

  • Gerard J. Chang;Ting-Pang Chang;Li-Da Tong

  • Affiliations:

  • Department of Mathematics, National Taiwan University, Taipei, Taiwan 10617 and Institute for Mathematical Sciences, National Taiwan University, Taipei, Taiwan 10617 and National Center for Theore ...;Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan 804;National Center for Theoretical Sciences, Shin-Chu, Taiwan and Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan 804

  • Venue:
  • Journal of Combinatorial Optimization
  • Year:
  • 2012

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Abstract

For the study of hamiltonicity of graphs and digraphs, Goodman and Hedetniemi introduced the concept of Hamiltonian number. The Hamiltonian number h(D) of a digraph D is the minimum length of a closed walk containing all vertices of D. In this paper, we study Hamiltonian numbers of the following proposed networks, which include strongly connected double loop networks. For integers d驴1, m驴1 and 驴驴0, the Möbius double loop network MDL(d,m,驴) is the digraph with vertex set {(i,j):0驴i驴d驴1,0驴j驴m驴1} and arc set {(i,j)(i+1,j) or (i,j)(i+1,j+1):0驴i驴d驴2,0驴j驴m驴1}驴{(d驴1,j)(0,j+驴) or (d驴1,j)(0,j+驴+1):0驴j驴m驴1}, where the second coordinate y of a vertex (x,y) is taken modulo m. We give an upper bound for the Hamiltonian number of a Möbius double loop network. We also give a necessary and sufficient condition for a Möbius double loop network MDL(d,m,驴) to have Hamiltonian number at most dm, dm+d, dm+1 or dm+2.