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On perfect Γ-decompositions of the complete graph

✍ Scribed by Marco Buratti; Anita Pasotti

Publisher: John Wiley and Sons
Year: 2009
Tongue: English
Weight: 155 KB
Volume: 17
Category: Article
ISSN: 1063-8539
DOI: 10.1002/jcd.20199

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✦ Synopsis

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 K~v~ to be i‐perfect if for every edge [x, y] of K~v~ 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 K~v~ 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 K~v~ 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

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