Perfect 1-Factorisations Of

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.

Cycles of quadratic Latin squares and anti-perfect $1$-factorisations

“…There are only three known infinite families [19,5] of perfect 1-factorisations of complete graphs. These families prove the existence of perfect 1-factorisations of K 2n where 2n ∈ {p + 1, 2p} for an odd prime p. Perfect 1-factorisations of K 2n are also known to exist for some sporadic values of n. See [15] for a list of these values.…”

“…An atomic Latin square is a Latin square whose conjugates are all row-Hamiltonian. Such squares have been studied in [5,34,27,15,36]. We define a Latin square of order n to be anti-atomic if none of its conjugates contain a row cycle of length n. We prove the following.…”

“…(ii) b = 2 − a and (q, a) ∈ {(13, 3), (13,8), (17,12), (17,15), (37,11), (37,27), (41, 13), (41, 25)}.…”

“…Such squares are called quadratic Latin squares. Quadratic Latin squares have previously been used to construct perfect 1-factorisations [34,15,1], mutually orthogonal Latin squares [14,13], atomic Latin squares [34], Falconer varieties [1], and maximally non-associative quasigroups [9,10]. Quadratic Latin squares are the main focus of this paper.…”

Cycles of quadratic Latin squares and antiperfect 1‐factorisations

“…The condition $$\{ab,(a-1)(b-1)\}\subseteq {{\rm{ {\mathcal R} }}}_{q}$$ ensures that $${\rm{ {\mathcal L} }}[a,b]$$ is a Latin square [16]. Quadratic Latin squares have previously been used to construct perfect 1‐factorisations [1, 17, 38], mutually orthogonal Latin squares [15, 16], atomic Latin squares [38], Falconer varieties [1] and maximally nonassociative quasigroups [10, 11]. Quadratic Latin squares are the main focus of this paper.…”