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- W3174596876 abstract "An avoidance problem of configurations in 4-cycle systems is investigated by generalizing the notion of sparseness, which is originally from ErdH{o}s' r-sparse conjecture on Steiner triple systems. A 4-cycle system of order v, 4CS(v), is said to be r-sparse if for every integer j satisfying 1 < j < r+1 it contains no configurations consisting of j 4-cycles whose union contains precisely j+3 vertices. If an r-sparse 4CS(v) is also free from copies of a configuration on two 4-cycles sharing a diagonal, called the double-diamond, we say it is strictly r-sparse. In this paper, we show that for every admissible order v there exists a strictly 4-sparse 4CS(v). We also prove that for any positive integer r > 1 and sufficiently large integer v there exists a constant number c such that there exists a strictly r-sparse 4-cycle packing of order v with cv^2 4-cycles." @default.
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- W3174596876 date "2012-09-20" @default.
- W3174596876 modified "2023-09-27" @default.
- W3174596876 title "Sparseness of 4-cycle systems" @default.
- W3174596876 doi "https://doi.org/10.48550/arxiv.1209.4438" @default.
- W3174596876 hasPublicationYear "2012" @default.
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