Low Weight Perfect Matchings

Answering a question posed by Caro, Hansberg, Lauri, and Zarb, we show that for every positive integer n and every function sigma: E(K-4n) -> {-1, 1} with sigma: E(K-4n) = 0, there is a perfect matching M in K(4n)with sigma(M) = 0. Strengthening the consequence of a result of Caro and Yuster, we show that for every positive integer n and every function sigma: E(K-4n) ->{-1, 1} with la vertical bar sigma(E(K-4n))vertical bar < n(2) + 11n+2, there is a perfect matching M in K-4n with vertical bar sigma(M)vertical bar <= 2. Both these results are best possible.