×

Embedding factorizations for 3-uniform hypergraphs II: \(r\)-factorizations into \(s\)-factorizations. (English) Zbl 1337.05084

Summary: Motivated by a 40-year-old problem due to P. J. Cameron [Parallelisms of complete designs. Cambridge etc.: Cambridge University Press (1976; Zbl 0333.05007)] on extending partial parallelisms, we provide necessary and sufficient conditions under which one can extend an \(r\)-factorization of a complete \(3\)-uniform hypergraph on \(m\) vertices, \(K_m^3\), to an \(s\)-factorization of \(K_n^3\). This generalizes an existing result of ZS. Baranyai and A. E. Brouwer [Extension of colourings of the edges of a complete (uniform hyper) graph. Amsterdam: Math. Centrum (1977; Zbl 0362.05059)] – where they proved it for the case \(r=s=1\).
For Part I see [A. Bahmanian and C. Rodger, J. Graph Theory 73, No. 1–2, 216–224 (2013; Zbl 1264.05088)].

MSC:

05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
05C65 Hypergraphs
05C15 Coloring of graphs and hypergraphs
05B40 Combinatorial aspects of packing and covering
05B05 Combinatorial aspects of block designs
PDFBibTeX XMLCite
Full Text: Link

References:

[1] L. D. Andersen and A. J. W. Hilton. Generalized Latin rectangles. II. Embedding. Discrete Math., 31(3):235-260, 1980. · Zbl 0476.05018
[2] M. A. Bahmanian. Detachments of hypergraphs I: The Berge-Johnson problem. Combin. Probab. Comput., 21(4):483-495, 2012. · Zbl 1247.05161
[3] M. A. Bahmanian and Mike Newman. Extending factorizations of complete uniform hypergraphs. Submitted. · Zbl 1337.05084
[4] M. A. Bahmanian and C. A. Rodger. Embedding factorizations for 3-uniform hypergraphs. J. Graph Theory, 73(2):216-224, 2013. · Zbl 1264.05088
[5] Zs. Baranyai. On the factorization of the complete uniform hypergraph. In Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erd˝os on his 60th birthday), Vol. I, pages 91-108. Colloq. Math. Soc. J´an¯os Bolyai, Vol. 10. North-Holland, Amsterdam, 1975. · Zbl 0306.05137
[6] Zs. Baranyai and A. E. Brouwer. Extension of colorings of the edges of a complete (uniform hyper)graph. Technical report, Mathematisch Centrum Amsterdam, Math. Centre Report ZW91, Zbl. 362.05059, 1977. · Zbl 0362.05059
[7] Peter J. Cameron. Parallelisms of complete designs. Cambridge University Press, Cambridge-New York-Melbourne, 1976. London Mathematical Society Lecture Note Series, No. 23. · Zbl 0333.05007
[8] Allan B. Cruse. On embedding incomplete symmetric Latin squares. J. Combin. Theory Ser. A, 16:18-22, 1974. · Zbl 0274.05016
[9] R. H¨aggkvist and T. Hellgren. Extensions of edge-colourings in hypergraphs. I. In Combinatorics, Paul Erd˝os is eighty, Vol. 1, Bolyai Soc. Math. Stud., pages 215-238. J´anos Bolyai Math. Soc., Budapest, 1993. · Zbl 0795.05053
[10] A. J. W. Hilton, Matthew Johnson, C. A. Rodger, and E. B. Wantland. Amalgamations of connected k-factorizations. J. Combin. Theory Ser. B, 88(2):267-279, 2003. the electronic journal of combinatorics 23(2) (2016), #P2.4213 · Zbl 1033.05084
[11] Matthew Johnson. Amalgamations of factorizations of complete graphs. J. Combin. Theory Ser. B, 97(4):597-611, 2007. · Zbl 1153.05055
[12] C. A. Rodger and E. B. Wantland. Embedding edge-colorings into 2-edge-connected k-factorizations of Kkn+1. J. Graph Theory, 19(2):169-185, 1995. · Zbl 0815.05050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.