A perfect one‐factorisation of

Pike, David A.

doi:10.1002/jcd.21653

Cited by 4 publications

(3 citation statements)

References 11 publications

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“…They cover all orders of the form n = p + 1 or n = 2p where p is an odd prime. Sporadic constructions including those reported in §5, together with [7,8,10,13], demonstrate existence for n 56 and the following orders: 126, 170, 244, 344, 530, 730, 1332, 1370, 1850, 2198, 2810, 3126, 4490, 6860, 6890, 11450, 11882, 12168, 15626, 16808, 22202, 24390, 24650, 26570, 29792, 29930, 32042, 38810, 44522, 50654, 51530, 52442, 63002, 72362, 76730, 78126, 79508, 103824, 148878, 161052, 205380, 226982, 300764, 357912, 371294, 493040, 571788, 1030302, 1092728, 1225044, 1295030, 2048384, 2248092, 2476100, 2685620, 3307950, 3442952, 4330748, 4657464, 5735340, 6436344, 6967872, 7880600, 9393932, 11089568, 11697084, 13651920, 15813252, 18191448, 19902512,…”

Section: Introductionmentioning

confidence: 98%

Perfect 1-factorisations of $K_{16}$

Gill,

Wanless

2019

Preprint

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We report the results of a computer enumeration that found that there are 3155 perfect 1factorisations (P1Fs) of the complete graph K 16 . Of these, 89 have a non-trivial automorphism group (correcting an earlier claim of 88 by Meszka and Rosa [6]).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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Section: Introductionmentioning

confidence: 98%

Perfect 1-factorisations of $K_{16}$

Gill,

Wanless

2019

Preprint

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“…They cover all orders of the form n = p + 1 or n = 2p, where p is an odd prime. Sporadic constructions including those reported in Section 5, together with [7,9,11,14], demonstrate existence for n ≤ 56 and the following orders: 126,170,244,344,530,730,1332,1370,1850,2198,2810,3126,4490,6860,6890,11450,11882,12168,15626,16808,22202,24390,24650,26570,29792,29930,32042,38810,44522,50654,51530,52442,63002,72362,76730,78126,79508,103824,148878,161052,205380,226982,300764,357912,371294,493040,571788,1030302,1092728,1225044,1295030,2048384,2248092,2476100,2685620,<...>…”

Section: Introductionmentioning

confidence: 99%

Perfect 1-Factorisations Of

Gill

Wanless²

2019

Bull. Aust. Math. Soc.

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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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“…The perfect one-factorization conjecture, posed by Kotzig [11], claims the existence of perfect one-factorizations for every even order ≥ n 2 4 of the complete graph K n 2 ; it still seems to be far from proven. In particular, the smallest order for which the conjecture remains unsettled is 64 [17]. For surveys on perfect one-factorizations, see [20,22].…”

Section: Introductionmentioning

confidence: 99%