Abstract
If the complete graph K n has vertex set X, a maximum packing of K n with 4-cycles, (X, C, L), is an edge-disjoint decomposition of K n into a collection C of 4-cycles so that the unused edges (the set L) is as small a set as possible. Maximum packings of K n with 4-cycles were shown to exist by Schönheim and Bialostocki (Can. Math. Bull. 18:703–708, 1975). An almost parallel class of a maximum packing (X, C, L) of K n with 4-cycles is a largest possible collection of vertex disjoint 4-cycles (so with \({\lfloor/4\rfloor}\) 4-cycles in it). In this paper, for all orders n, except 9, which does not exist, and possibly 23, 41 and 57, we exhibit a maximum packing of K n with 4-cycles so that the 4-cycles in the packing are resolvable into almost parallel classes, with any remaining 4-cycles being vertex disjoint. [Note: The three missing orders have now been found, and appear in Billington et al. (to appear).]
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Billington, E.J., Hoffman, D.G., Lindner, C.C., Meszka, M.: Almost resolvable minimum coverings of complete graphs with 4-cycles (to appear)
Colbourn C.J., Hoffman D.G., Rees R.: A new class of group divisible designs with block size three. J. Combin. Theory Ser. A 59, 73–89 (1992)
Dejter I.J., Lindner C.C., Meszka M., Rodger C.A.: Almost resolvable 4-cycle systems. J. Combin. Math. Combin. Computing 63, 173–182 (2007) [Corrigendum/Addendum ibid. 66, 297–298 (2008)]
Lindner C.C., Rodger C.A.: Design Theory, 2nd edn, pp. 264. CRC Press, Boca Raton (2009)
Schönheim J., Bialostocki A.: Packing and covering the complete graph with 4-cycles. Can. Math. Bull. 18, 703–708 (1975)
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Billington, E.J., Dejter, I.J., Hoffman, D.G. et al. Almost Resolvable Maximum Packings of Complete Graphs with 4-Cycles. Graphs and Combinatorics 27, 161–170 (2011). https://doi.org/10.1007/s00373-010-0967-0
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DOI: https://doi.org/10.1007/s00373-010-0967-0