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Decompositions of complete graphs into factors with diameter two. (English) Zbl 0213.51003
Let $$f(k)$$ be the smallest integer $$n$$ so that the complete graph $$\langle n \rangle$$ can be decomposed into $$k$$ edge disjoint subgraphs of diameter $$2$$. The authors prove: (1) $$f(k)<\left({49 \over 10} \right)^2k^2 \log k$$. Recently N. Sauer considerably improved (1) by showing $$f(k)<7k$$ [cf. the paper reviewed below, J. Comb. Theory 9, 423-426 (1970; Zbl 0213.51004)].

##### MSC:
 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
##### References:
 [1] Bosák J., Rosa A., Znám Š.: On decompositions of complete graphs into factors with given diameters. Theory of Graphs, Proc. Colloq. Tihany 1966, Akadémiai Kiadó, Budapest 1968, 37 - 56. · Zbl 0159.54203 [2] Erdös P., Rényi A.: On the evolution of random graphs. Magyar tud. akad. Mat. kutató int. közl. 5 (1960), 17 61. [3] Erdös P., Rényi A.: Additive properties of random sequences of positive integers. Acta Arithm. 6 (1960), 83-110. · Zbl 0091.04401 [4] Erdös P., Rényi A.: Egg gráfelméleti problémáról. Magyar tud. akad. Mat. kutató int. közl. 7 (1962), 623-641. [5] Виноградов И. М.: Основы мєоруу чусєл. ГИТТЛ, Москва-Лєнинград 1952.
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