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PERFECT 1-FACTORISATIONS OF $K_{16}$

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

Abstract

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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[3] Dinitz, J. H. and Garnick, D. K., ‘There are 23 non-isomorphic perfect one-factorisations of K 14 ’, J. Combin. Des. 4 (1996), 14.
[4] Kaski, P., Medeiros, A. D. S., Östergård, P. R. J. and Wanless, I. M., ‘Switching in one-factorisations of complete graphs’, Electron. J. Combin. 21(2) (2014), #P2.49.
[5] Meszka, M., ‘There are 3155 nonisomorphic perfect 1-factorizations of $K_{16}$ ’, submitted for publication.
[6] Meszka, M. and Rosa, A., ‘Perfect 1-factorizations of K 16 with nontrivial automorphism group’, J. Combin. Math. Combin. Comput. 47 (2003), 97111.
[7] Pike, D. A., ‘A perfect 1-factorisation of K 56 ’, J. Combin. Des. 27 (2019), 386390.
[8] Rosa, A., ‘Perfect 1-factorizations’, Math. Slovaca 69 (2019), 479496.
[9] Seah, E., ‘Perfect one-factorizations of the complete graph—a survey’, Bull. Inst. Combin. Appl. 1 (1991), 5970.
[10] Wallis, W. D., One-factorizations (Kluwer Academic, Dordrecht, Netherlands, 1997).
[11] Wanless, I. M., ‘Atomic Latin squares based on cyclotomic orthomorphisms’, Electron. J. Combin. 12 (2005), #R22.
[12] Wanless, I. M., Author’s homepage, http://users.monash.edu.au/∼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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