New families of atomic Latin squares and perfect 1-factorisations

Darryn Bryant*, Barbara Maenhaut, Ian M. Wanless

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    24 Citations (Scopus)


    A perfect 1 -factorisation of a graph G is a decomposition of G into edge disjoint 1 -factors such that the union of any two of the factors is a Hamiltonian cycle. Let p ≥ 11 be prime. We demonstrate the existence of two non-isomorphic perfect 1-factorisations of Kp+1 (one of which is well known) and five non-isomorphic perfect 1-factorisations of Kp,p. If 2 is a primitive root modulo p, then we show the existence of 11 non-isomorphic perfect 1-factorisations of Kp,p and 5 main classes of atomic Latin squares of order p. Only three of these main classes were previously known. One of the two new main classes has a trivial autotopy group.

    Original languageEnglish
    Pages (from-to)608-624
    Number of pages17
    JournalJournal of Combinatorial Theory. Series A
    Issue number4
    Publication statusPublished - May 2006


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