Resolvable path designs

Abstract

AbstractA resolvable (balanced) path design, RBPD(v, k, λ) is the decomposition of λ copies of the complete graph on v vertices into edge-disjoint subgraphs such that each subgraph consists of vk vertex-disjoint paths of length k − 1 (k vertices). It is shown that an RBPD(v, 3, λ) exists if and only if v ≡ 9 (modulo 12/gcd(4, λ)). Moreover, the RBPD(v, 3, λ) can have an automorphism of order v3. For k > 3, it is shown that if v is large enough, then an RBPD(v, k, 1) exists if and only if v ≡ k2 (modulo lcm(2k − 2, k)). Also, it is shown that the categorical product of a k-factorable graph and a regular graph is also k-factorable. These results are stronger than two conjectures of P. Hell and A. Ros