Geodesic
Decomposition of Complete Multigraphs Into Stars and Cycles
Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 4, pp. 629-639

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Let k be a positive integer, Sk and Ck denote, respectively, a star and a cycle of k edges. λKn is the usual notation for the complete multigraph on n vertices and in which every edge is taken λ times. In this paper, we investigate necessary and sufficient conditions for the existence of the decomposition of λKn into edges disjoint of stars Sk’s and cycles Ck’s.
Keywords: graph decomposition, complete multigraph, stars, cycles 
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Beggas, Fairouz; Haddad, Mohammed; Kheddouci, Hamamache. Decomposition of Complete Multigraphs Into Stars and Cycles. Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 4, pp. 629-639. http://geodesic.mathdoc.fr/item/DMGT_2015_35_4_a2/