Decomposition of Complete Graphs into Isomorphic Factors with a Given Diameter

The study of decompositions of the complete graph Kn into factors with given diameters was initiated in [4]. In [4], the function F(du d2, ..., dm) was defined to be the smallest cardinal number n (if it exists) such that Kn can be decomposed into m factors with diameters dt,d2, • •-,dm; if such an n does not exist then F(di,d2, ..., dm) =oo . It was then shown (Theorem 1 in [4]) that Kn can be decomposed into n factors with the diameters dud2, '•-,dm if and only if n ^ F(d1,d2, ...,dm). A fairly large number of papers [2, 3, 4, 7, 8, 10] deal with the determination of the values of the function F{dud2, ...,dm), while several other papers study analogous problems and functions for the case of directed graphs [11, 13, 14], 2-graphs [1] and bipartite graphs [12]. The papers [2, 3, 4, 7, 8,10] are devoted, in particular, to the case of equal diameters, i.e. dy = d2 = ... = dm (in this case one writes ^ d ) = Fm(d)).