The existence of restricted resolvable designs I: (1,2)-factorizations of K2 n

Abstract

It has been known for at least a century that (in modern terminology) the complete graph K 2 n admits a 1-factorization, that is, a partition of its edge set E into subsets E 1,…, E 2 n−1 such that each E i consists of n vertex-disjoint edges. A considerably newer result (due to Ray-Chaudhuri and Wilson) states that if n is an odd integer then the complete graph K 3 n admits what we will call a 2-factorization, that being a pair ( T, P) where T is a decomposition of K 3 n into triangles ( K 3's) and P is a partition of T into subsets T 1,…, T (3n−1) 2 so that each T i consists of n vertex-disjoint triangles. Between these two extremes we define a (1, 2)- factorization of K n with cardinality k to be a pair ( T, P) where T is a decomposition of K n into edges and triangles ( K 2's and K 3's) and P is a partition of T into subsets T 1,…, T k such that each T i is a vertex partition of K n . This is the first in a series of two papers in which we investigate the following question: for which integers n > 0 and ⌊ n/2⌋ ⩽ k ⩽ n − 1 does the complete graph K n admit a (1, 2)-factorization of cardinality k? We prove here that when n is even the ‘obvious’ necessary conditions for the existence of these designs are sufficient, with exactly two exceptions: n = 6, k = 3; and n = 12, k = 6.