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  • MICHAEL J. GILL (a1) and IAN M. WANLESS (a2)


We report the results of a computer enumeration that found that there are 3155 perfect 1-factorisations (P1Fs) of the complete graph  $K_{16}$ . Of these, 89 have a nontrivial automorphism group (correcting an earlier claim of 88 by Meszka and Rosa [‘Perfect 1-factorisations of $K_{16}$ with nontrivial automorphism group’, J. Combin. Math. Combin. Comput. 47 (2003), 97–111]). We also (i) describe a new invariant which distinguishes between the P1Fs of  $K_{16}$ , (ii) observe that the new P1Fs produce no atomic Latin squares of order 15 and (iii) record P1Fs for a number of large orders that exceed prime powers by one.


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Research supported by an Australian Government Research Training Program (RTP) Scholarship and by ARC grant DP150100506.



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[1] Bryant, D., Maenhaut, B. and Wanless, I. M., ‘New families of atomic Latin squares and perfect one-factorisations’, J. Combin. Theory Ser. A 113 (2006), 608624.
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[6] Meszka, M. and Rosa, A., ‘Perfect 1-factorizations of K 16 with nontrivial automorphism group’, J. Combin. Math. Combin. Comput. 47 (2003), 97111.
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[8] Rosa, A., ‘Perfect 1-factorizations’, Math. Slovaca 69 (2019), 479496.
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[12] Wanless, I. M., Author’s homepage,∼iwanless/data/P1F/newP1F.html.
[13] Wanless, I. M. and Ihrig, E. C., ‘Symmetries that Latin squares inherit from 1-factorizations’, J. Combin. Des. 13 (2005), 157172.
[14] Wolfe, A. J., ‘A perfect one-factorization of K 52 ’, J. Combin. Des. 17 (2009), 190196.
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