| 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 1‐factor of a latin square 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 1‐factors with exactly n ? 1 distinct symbols is even. Also we prove that if the complete graph K2n, n ≥ 8, is edge colored such that each color appears on at most  edges, then there exists a multicolored perfect matching. © 2004 Wiley Periodicals, Inc. |
