Abstract
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.
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Parker, A., Rodger, C.A. Embedding Partial 4-cycle Systems of Arbitrary Index. Graphs and Combinatorics 24, 367–371 (2008). https://doi.org/10.1007/s00373-008-0791-y
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DOI: https://doi.org/10.1007/s00373-008-0791-y