Geodesic
Decomposition of complete graphs into factors of diameter two and three
Discussiones Mathematicae. Graph Theory, Tome 23 (2003) no. 1, pp. 37-54

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We analyze a minimum number of vertices of a complete graph that can be decomposed into one factor of diameter 2 and k factors of diameter at most 3. We find exact values for k ≤ 4 and the asymptotic value of the ratio of this number and k when k tends to infinity. We also find the asymptotic value of the ratio of the number of vertices of the smallest complete graph that can be decomposed into p factors of diameter 2 and k factors of diameter 3 and number k when p is fixed.
Keywords: decomposition, graph 
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Vukicević, Damir. Decomposition of complete graphs into factors of diameter two and three. Discussiones Mathematicae. Graph Theory, Tome 23 (2003) no. 1, pp. 37-54. http://geodesic.mathdoc.fr/item/DMGT_2003_23_1_a2/