Geodesic
Symmetric Hamilton Cycle Decompositions of Complete Multigraphs
Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 4, pp. 695-707.

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Let n ≥ 3 and ⋋ ≥ 1 be integers. Let ⋋K_n denote the complete multigraph with edge-multiplicity ⋋. In this paper, we show that there exists a symmetric Hamilton cycle decomposition of ⋋K_2m for all even ⋋ ≥ 2 and m ≥ 2. Also we show that there exists a symmetric Hamilton cycle decomposition of ⋋K_2m − F for all odd ⋋ ≥ 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 ⋋K_n (respectively, ⋋K_n − F, where F is a 1-factor of ⋋K_n) which exist if and only if ⋋(n − 1) is even (respectively, ⋋(n − 1) is odd), except the non-existence cases n ≡ 0 or 6 (mod 8) when ⋋ = 1
Keywords: complete multigraph, 1-factor, symmetric Hamilton cycle, decomposition 
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Chitra, V.; Muthusamy, A. Symmetric Hamilton Cycle Decompositions of Complete Multigraphs. Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 4, pp. 695-707. http://geodesic.mathdoc.fr/item/DMGT_2013_33_4_a5/

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