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- W2018465533 abstract "Generalizing the well-known concept of an i-perfect cycle system, Pasotti [Pasotti, in press, Australas J Combin] defined a Γ-decomposition (Γ-factorization) of a complete graph Kv to be i-perfect if for every edge [x, y] of Kv there is exactly one block of the decomposition (factor of the factorization) in which x and y have distance i. In particular, a Γ-decomposition (Γ-factorization) of Kv that is i-perfect for any i not exceeding the diameter of a connected graph Γ will be said a Steiner (Kirkman) Γ-system of order v. In this article we first observe that as a consequence of the deep theory on decompositions of edge-colored graphs developed by Lamken and Wilson [Lamken and Wilson, 2000, J Combin Theory Ser A 89, 149–200], there are infinitely many values of v for which there exists an i-perfect Γ-decomposition of Kv provided that Γ is an i-equidistance graph, namely a graph such that the number of pairs of vertices at distance i is equal to the number of its edges. Then we give some concrete direct constructions for elementary abelian Steiner Γ-systems with Γ the wheel on 8 vertices or a circulant graph, and for elementary abelian 2-perfect cube-factorizations. We also present some recursive constructions and some results on 2-transitive Kirkman Γ-systems. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 197–209, 2009" @default.
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- W2018465533 date "2009-03-01" @default.
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- W2018465533 title "On perfect Γ-decompositions of the complete graph" @default.
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- W2018465533 doi "https://doi.org/10.1002/jcd.20199" @default.
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