Discussiones Mathematicae Graph Theory 33(4) (2013) 695-707
doi: 10.7151/dmgt.1687

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Symmetric Hamilton Cycle Decompositions of Complete Multigraphs

V. Chitra and A. Muthusamy

Department of Mathematics
Periyar University
Salem-636 011, TN, India

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.

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Received 16 June 2011
Revised 14 August 2012
Accepted 20 August 2012