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Factorizations of complete multipartite hypergraphs. (English) Zbl 1351.05160
Summary: In a mathematics workshop with \(m n\) mathematicians from \(n\) different areas, each area consisting of \(m\) mathematicians, we want to create a collaboration network. For this purpose, we would like to schedule daily meetings between groups of size three, so that (i) two people of the same area meet one person of another area, (ii) each person has exactly \(r\) meeting(s) each day, and (iii) each pair of people of the same area have exactly \(\lambda\) meeting(s) with each person of another area by the end of the workshop. Using hypergraph amalgamation-detachment, we prove a more general theorem. In particular we show that above meetings can be scheduled if: \(3 \mid r m\), \(2 \mid rnm\) and \(r \mid 3\lambda(n-1) \binom{m}{2}\). This result can be viewed as an analogue of Baranyai’s theorem on factorizations of complete multipartite hypergraphs.
MSC:
05C65 Hypergraphs
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
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[1] Bahmanian, M. A., Detachments of hypergraphs I: the berge-Johnson problem, Combin. Probab. Comput., 21, 4, 483-495, (2012) · Zbl 1247.05161
[2] Bahmanian, M. A., Detachments of amalgamated 3-uniform hypergraphs factorization consequences, J. Combin. Designs, 20, 527-549, (2012) · Zbl 1258.05087
[3] Bahmanian, M. A., Connected baranyai’s theorem, Combinatorica, 34, 2, 129-138, (2014) · Zbl 1324.05053
[4] Bahmanian, M. A.; Newman, Mike, Embedding factorizations for 3-uniform hypergraphs II: \(r\)-factorizations into \(s\)-factorizations, Electron. J. Combin., 23, 2, (2016), Paper 2.42, 14 pp · Zbl 1337.05084
[5] Bahmanian, M. A.; Rodger, C. A., Multiply balanced edge colorings of multigraphs, J. Graph Theory, 70, 3, 297-317, (2012) · Zbl 1244.05085
[6] Bahmanian, M. A.; Rodger, C. A., (What are Graph Amalgamations? Recent Results in Designs and Graphs: a Tribute to Lucia Gionfriddo, Quaderni di matematica, vol. 28, (2013), University of Naples), 63-82
[7] Bahmanian, M. A.; Rodger, C. A., Embedding factorizations for 3-uniform hypergraphs, J. Graph Theory, 73, 2, 216-224, (2013) · Zbl 1264.05088
[8] Baranyai, Zs., (On The Factorization of The Complete Uniform Hypergraph, Infinite and Finite Sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. I,, Colloq. Math. Soc. Janos Bolyai, vol. 10, (1975), North-Holland Amsterdam), 91-108
[9] Baranyai, Zs., The edge-coloring of complete hypergraphs I, J. Combin. Theory B, 26, 3, 276-294, (1979) · Zbl 0413.05040
[10] Hilton, A. J.W., Hamilton decompositions of complete graphs, J. Combin. Theory B, 36, 125-134, (1984) · Zbl 0542.05044
[11] Hilton, A. J.W.; Rodger, C. A., Hamilton decompositions of complete regular \(s\)-partite graphs, Discrete Math., 58, 63-78, (1986) · Zbl 0593.05047
[12] Johnson, M., Amalgamations of factorizations of complete graphs, J. Combin. Theory Ser. B, 97, 4, 597-611, (2007) · Zbl 1153.05055
[13] Johnstone, W. R., Decompositions of complete graphs, Bull. London Math. Soc., 32, 2, 141-145, (2000) · Zbl 1020.05056
[14] Kirkman, T. P, On a problem in combinations, Camb. Dublin Math. J., 2, 191-204, (1847)
[15] Nash-Williams, C. St. J.A., Amalgamations of almost regular edge-colourings of simple graphs, J. Combin. Theory B, 43, 322-342, (1987) · Zbl 0654.05031
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