“School of Mathematics”
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Paper IPM / M / 7292  


Abstract:  
Ryser conjectured that the number of transversals of a latin
square of order n is congruent to n modulo 2. Balasubramanian
has shown that the number of transversals of a latin square of
even order is even. A 1factor of a latin of order n is a set of
n cells no two from the same row or the same column. We prove
that for any latin square of order n, the number of 1factors
with exactly n−1 distinct symbols is even. Also we prove that if
the complete graph K_{2n}, n ≥ 8, is edge colored such that
each color appears on at most [(n−2)/(2e)] edges, then there
exists a multicolored perfect matching.
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