The existence of restricted resolvable designs. I: (1,2)-factorizations of \(K_{2n}\)
A (1,2)-factorization of the complete graph \(K_ n\) is a partition P of the edges of \(K_ n\) into edges and triangles, and a partition T of P into k subsets, called (1,2)-factors, each of which partitions the vertices of \(K_ n\). When all members of P are edges, this is a one- factorization; when all members are triangles, this is a Kirkman triple system. The author addresses the question: for which integers \(n>0\) and \(\lfloor n/2\rfloor \leq k\leq n\) does \(K_ n\) admit a (1,2)-factorization with k (1,2)-factors. Part I settles this problem completely for n even, showing that except when \(n=6\), \(k=3\) or \(n=12\), \(k=6\), such a (1,2)- factorization exists provided the obvious arithmetic conditions are met. The proof is recursive, using powerful methods involving frames. Part II treats the apparently more complicated case when n is odd (cf. below). Here in addition to obvious arithmetic conditions, one requires that one of the following holds: 1. \(n\equiv 3(mod 6)\) and \(k=(n-1)/2\); 2. \(n=9\) and \(k=6\); or 3. \((n+1)/2\leq k\leq n-4\). The author settles the existence of (1,2)-factorizations for odd n, leaving eighteen orders unsettled (the largest being 89). The author [``The spectrum of restricted resolvable designs with \(r=2''\), Discrete Math., to appear] has since completed the determination for all orders.
scientific article; zbMATH DE number 4150165
scientific article