Embedding Partial 4-cycle Systems of Arbitrary Index

摘要

Recently, Lindner showed that every partial 4-cycle system of order n and index 1 could be embedded in a 4-cycle system of order ν and index 1 with v ≤ 2n + 15. While the technique he used does not immediately extend to any higher index, here we develop his ideas to show that every partial 4-cycle system of order n and index λ can be embedded in a 4-cycle system of order v and index λ for all λ-admissible $nu geq 8 lceil n /4 rceil + 7, nu ne 8 lceil n /4 rceil + 8$ . This improves on the best known bounds of v = 4n and v = 8n + 1 when λ > 1 is even and odd respectively.