New families of atomic Latin squares and perfect 1-factorisations

Darryn Bryant; Barbara Maenhaut; Ian M. Wanless

doi:10.1016/j.jcta.2005.05.003

New families of atomic Latin squares and perfect 1-factorisations

Darryn Bryant^*, Barbara Maenhaut, Ian M. Wanless

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

27 Citations (Scopus)

Abstract

A perfect 1 -factorisation of a graph G is a decomposition of G into edge disjoint 1 -factors such that the union of any two of the factors is a Hamiltonian cycle. Let p ≥ 11 be prime. We demonstrate the existence of two non-isomorphic perfect 1-factorisations of K_p+1 (one of which is well known) and five non-isomorphic perfect 1-factorisations of K_p,p. If 2 is a primitive root modulo p, then we show the existence of 11 non-isomorphic perfect 1-factorisations of K_p,p and 5 main classes of atomic Latin squares of order p. Only three of these main classes were previously known. One of the two new main classes has a trivial autotopy group.

Original language	English
Pages (from-to)	608-624
Number of pages	17
Journal	Journal of Combinatorial Theory. Series A
Volume	113
Issue number	4
DOIs	https://doi.org/10.1016/j.jcta.2005.05.003
Publication status	Published - May 2006

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title = "New families of atomic Latin squares and perfect 1-factorisations",

abstract = "A perfect 1 -factorisation of a graph G is a decomposition of G into edge disjoint 1 -factors such that the union of any two of the factors is a Hamiltonian cycle. Let p ≥ 11 be prime. We demonstrate the existence of two non-isomorphic perfect 1-factorisations of Kp+1 (one of which is well known) and five non-isomorphic perfect 1-factorisations of Kp,p. If 2 is a primitive root modulo p, then we show the existence of 11 non-isomorphic perfect 1-factorisations of Kp,p and 5 main classes of atomic Latin squares of order p. Only three of these main classes were previously known. One of the two new main classes has a trivial autotopy group.",

keywords = "Atomic latin square, Autotopy group, Even starter, Hamiltonian cycle, Perfect 1-factorisation, Totally symmetric",

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year = "2006",

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journal = "Journal of Combinatorial Theory. Series A",

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T1 - New families of atomic Latin squares and perfect 1-factorisations

AU - Bryant, Darryn

AU - Maenhaut, Barbara

AU - Wanless, Ian M.

PY - 2006/5

Y1 - 2006/5

N2 - A perfect 1 -factorisation of a graph G is a decomposition of G into edge disjoint 1 -factors such that the union of any two of the factors is a Hamiltonian cycle. Let p ≥ 11 be prime. We demonstrate the existence of two non-isomorphic perfect 1-factorisations of Kp+1 (one of which is well known) and five non-isomorphic perfect 1-factorisations of Kp,p. If 2 is a primitive root modulo p, then we show the existence of 11 non-isomorphic perfect 1-factorisations of Kp,p and 5 main classes of atomic Latin squares of order p. Only three of these main classes were previously known. One of the two new main classes has a trivial autotopy group.

AB - A perfect 1 -factorisation of a graph G is a decomposition of G into edge disjoint 1 -factors such that the union of any two of the factors is a Hamiltonian cycle. Let p ≥ 11 be prime. We demonstrate the existence of two non-isomorphic perfect 1-factorisations of Kp+1 (one of which is well known) and five non-isomorphic perfect 1-factorisations of Kp,p. If 2 is a primitive root modulo p, then we show the existence of 11 non-isomorphic perfect 1-factorisations of Kp,p and 5 main classes of atomic Latin squares of order p. Only three of these main classes were previously known. One of the two new main classes has a trivial autotopy group.

KW - Atomic latin square

KW - Autotopy group

KW - Even starter

KW - Hamiltonian cycle

KW - Perfect 1-factorisation

KW - Totally symmetric

UR - https://www.scopus.com/pages/publications/33645542951

U2 - 10.1016/j.jcta.2005.05.003

DO - 10.1016/j.jcta.2005.05.003

M3 - Article

SN - 0097-3165

VL - 113

SP - 608

EP - 624

JO - Journal of Combinatorial Theory. Series A

JF - Journal of Combinatorial Theory. Series A

IS - 4

ER -

New families of atomic Latin squares and perfect 1-factorisations

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