Small embeddings for partial 5‐cycle systems

Abstract

Abstract:It has been conjectured that any partial 5-cycle system of order u can be embedded in a 5-cycle system of order v whenever v ≥ 3u/2+1 and v ≡ 1, 5 (mod 10). The smallest known embeddings for any partial 5-cycle system of order u is 10u+5. In this paper we significantly improve this result by proving that for any partial 5-cycle system of order u ≥ 255, there exists a 5-cycle system of order at most (9u+146)/4 into which the partial 5-cycle system of order u can be embedded. q