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Discussiones Mathematicae Graph Theory 33(4) (2013)
695-707
DOI: https://doi.org/10.7151/dmgt.1687
Symmetric Hamilton Cycle Decompositions of Complete Multigraphs
| V. Chitra and A. Muthusamy
Department of Mathematics |
Abstract
Let n ≥ 3 and λ ≥ 1 be integers. Let λKn denote the complete multigraph with edge-multiplicity λ. In this paper, we show that there exists a symmetric Hamilton cycle decomposition of λK2m for all even λ ≥ 2 and m ≥ 2. Also we show that there exists a symmetric Hamilton cycle decomposition of λK2m −F for all odd λ ≥ 3 and m ≥ 2. In fact, our results together with the earlier results (by Walecki and Brualdi and Schroeder) completely settle the existence of symmetric Hamilton cycle decomposition of λKn (respectively, λKn −F, where F is a 1-factor of λKn) which exist if and only if λ(n −1) is even (respectively, λ(n −1) is odd), except the non-existence cases n ≡ 0 or 6 mod 8 when λ = 1.
Keywords: complete multigraph, 1-factor, symmetric Hamilton cycle, decomposition
2010 Mathematics Subject Classification: 05C45, 05C51, 05C70.
References
| [1] | J. Akiyama, M. Kobayashi and G. Nakamura, Symmetric Hamilton cycle decompositions of the complete graph, J. Combin. Des. 12 (2004) 39--45, doi: 10.1002/jcd.10066. |
| [2] | B. Alspach, The wonderful Walecki construction, Bull. Inst. Combin. Appl. 52 (2008) 7--20. |
| [3] | J. Bosák, Decompositions of Graphs (Kluwer Academic Publishers, 1990). |
| [4] | R.A. Brualdi and M.W. Schroeder, Symmetric Hamilton cycle decompositions of complete graphs minus a 1-factor, J. Combin. Des. 19 (2011) 1--15, doi: 10.1002/jcd.20257. |
| [5] | M. Buratti, S. Capparelli and A. Del Fra, Cyclic Hamiltonian cycle systems of the λ-fold complete and cocktail party graph, European J. Combin. 31 (2010) 1484--1496, doi: 10.1016/j.ejc.2010.01.004. |
| [6] | M. Buratti and A. Del Fra, Cyclic Hamiltonian cycle systems of the complete graph, Discrete Math. 279 (2004) 107--119, doi: 10.1016/S0012-365X(03)00267-X. |
| [7] | M. Buratti and F. Merola, Dihedral Hamiltonian cycle system of the cocktail party graph, J. Combin. Des. 21 (2013) 1--23, doi: 10.1002/jcd.21311. |
| [8] | A.J.W. Hilton, Hamiltonian decompositions of complete graphs, J. Combin. Theory (B) 36 (1984) 125--134, doi: 10.1016/0095-8956(84)90020-0. |
| [9] | H. Jordon and J. Morris, Cyclic hamiltonian cycle systems of the complete graph minus a 1-factor, Discrete Math. 308 (2008) 2440--2449, doi: 10.1016/j.disc.2007.05.009. |
| [10] | D.E. Lucas , Recreations Mathematiques, Vol.2 (Gauthiers Villars, Paris, 1982). |
Received 16 June 2011
Revised 14 August 2012
Accepted 20 August 2012
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