 
Summary: ACYCLIC MATCHINGS
NOGA ALON, C. KENNETH FAN, DANIEL KLEITMAN, AND JOZSEF LOSONCZY
Abstract. The purpose of this note is to give a constuctive proof of a conjecture
in [1] concerning the existence of acyclic matchings.
1. Main result
Let B, D Zn
. Assume that B = D and 0 D. A matching is a bijection
f : B D such that b + f(b) B for all b B. For any matching f, define
mf : Zn
Z by mf (v) = #{b B  b + f(b) = v}. An acyclic matching is a
matching f such that for any matching g such that mf = mg, we have f = g.
Theorem 1. There exists an acyclic matching.
This was first conjectured in [1]. The conjecure arises in the study of the problem,
considered by Wakeford [2], of deciding which sets of monomials are removable from a
generic homogeneous polynomial using a linear change of variables. For more details,
see [1]. The following proof is constructive.
Proof. First totally order Zn
so that if v > w then for any u, v + u > w + u (and
hence for v > 0, v + u > u.) For instance, choose a basis and order lexicographically.
Label the set B so that b1 < b2 < b3 < · · · < bm.
