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Decompositions of complete graphs into blown-up cycles \(C_m\)[2]. (English) Zbl 1230.05234

Summary: Let \(C_m[\overline K_2]\) stand for a cycle \(C_m\) in which every vertex is replaced by two isolated vertices and every edge by \(K_{2,2}\). We prove that the complete graph \(K_{8mk+1}\) can be decomposed into graphs isomorphic to \(C_m [\overline K_2]\) for any \(m\geq 3, k>0\). Decompositions of complete graphs into certain collections of even cycles are obtained as a corollary. Also some special cases of Alspach Conjecture are solved in this article. All proofs are constructive and use both graph theory and design theory techniques.

MSC:

05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
05C78 Graph labelling (graceful graphs, bandwidth, etc.)
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