On the completion of latin rectangles to symmetric latin squares

J. Aust. Math. Soc. 76 (2004), 109-124

On the completion of latin rectangles to symmetric latin squares

Darryn Bryant
Department of Mathematics
University of Queensland
Qld 4072
Australia
db@maths.uq.edu.au

and

C. A. Rodger
School of Mathematical and
Physical Sciences
University of Newcastle
NSW 2308
Australia
rodgec1@auburn.edu

Abstract

We find necessary and sufficient conditions for completing an arbitrary 2 by n latin rectangle to an n by n symmetric latin square, for completing an arbitrary 2 by n latin rectangle to an n by n unipotent symmetric latin square, and for completing an arbitrary 1 by n latin rectangle to an n by n idempotent symmetric latin square. Equivalently, we prove necessary and sufficient conditions for the existence of an (n - 1)-edge colouring of

(n even), and for an n -edge colouring of

(n odd) in which the colours assigned to the edges incident with two vertices are specified in advance.

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