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.
Original language | English |
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Pages (from-to) | 608-624 |
Number of pages | 17 |
Journal | Journal of Combinatorial Theory. Series A |
Volume | 113 |
Issue number | 4 |
DOIs | |
Publication status | Published - May 2006 |