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PERFECT 1-FACTORISATIONS OF $K_{16}$

Part of: Designs and configurations Graph theory

Published online by Cambridge University Press:  07 August 2019

MICHAEL J. GILL Open the ORCID record for MICHAEL J. GILL [Opens in a new window]  and
IAN M. WANLESS Open the ORCID record for IAN M. WANLESS [Opens in a new window]
MICHAEL J. GILL
Affiliation:
School of Mathematics, Monash University, Vic 3800, Australia email michael.gill@monash.edu
IAN M. WANLESS*
Affiliation:
School of Mathematics, Monash University, Vic 3800, Australia email ian.wanless@monash.edu
*
✉ian.wanless@monash.edu
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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.

Keywords

perfect 1-factorisationHamilton cycleatomic Latin squaretrainquotient coset starter

MSC classification

Primary: 05C70: Factorization, matching, partitioning, covering and packing
Secondary: 05B15: Orthogonal arrays, Latin squares, Room squares
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 101 , Issue 2 , April 2020 , pp. 177 - 185
Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

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

References

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