Row complete squares and a problem of A. Kotzig concerning P-quasigroups and eulerian circuits

Abstract

An n × n square L on n symbols is called row (column) complete if every ordered pair of the symbols of L occurs just once as an adjacent pair of elements in some row (column) of L. It is called row (column) latin if each symbol occurs exactly once in each row (column) of the square. A square which is both row latin and column latin is called a latin square. All known examples of row complete latin squares can be made column complete as well by suitable reordering of their rows and in the present paper we provide a sufficient condition that a given row complete latin square should have this property. Using row complete and column latin squares as a tool we follow this by showing how to construct code words on n symbols of the maximum possible length l = 1 2 n(n − 1) + 1 with the two properties that ( i) no unordered pair of consecutive symbols is repeated more than once and (ii) no unordered pair of nearly consecutive symbols is repeated more than once. (Two symbols are said to be nearly consecutive if they are separated by a single symbol.) We prove that such code words exist whenever n = 4 r + 3 with r ≢ 1 mod 6 and r ≢ 2 mod 5. We show that the existence of such a code word for a given value of n guarantees the existence of an Eulerian circuit in the complete undirected n-graph which corresponds to a P-quasigroup, thus answering a question raised by A. Kotzig in the affirmative. (Kotzig has defined a P-groupoid as a groupoid ( G, ·) having the following three properties: (i) a . a = a for all a ϵ G; (ii) a ≠ b implies a ≠ a . b and b ≠ a. b for all a, b ϵ G; and (iii) a . b = c implies c. b = a for all a, b, c ϵ G. Every decomposition of the complete undirected n-graph into disjoint closed circuits defines such a P-groupoid, as is easily seen by defining a . b = c if and only if a, b, c are consecutive edges of one such closed circuit. A P-groupoid whose multiplication table is a latin square is called a P-quasigroup.)