Acyclic Matchings

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Advances in Mathematics � AI1573

advances in mathematics 122, 234�236 (1996)

Acyclic Matchings

Noga Alon*, -

Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences,Tel Aviv University, Tel Aviv, Israel

C. Kenneth Fan� , 9

Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138

and

Daniel Kleitman*, � and Jozsef Losonczy&

Department of Mathematics, Massachusetts Institute of Technology,Cambridge, Massachusetts 02139

Received February 10, 1996

The purpose of this note is to give a constructive proof of a conjecture in Fanand Losonczy (1996) concerning the existence of acyclic matchings. � 1996 Academic

Press, Inc.

1. MAIN RESULT

Let B, D/Zn. Assume that |B|=|D| and 0 � D. A matching is a bijectionf : B � D such that b+ f (b) � B for all b # B. For any matching f, definemf : Zn � Z by mf (v)=*[b # B | b+ f (b)=v]. An acyclic matching is amatching f such that for any matching g such that mf =mg , we have f = g.

Theorem 1. There exists an acyclic matching.

article no. 0060

2340001-8708�96 �18.00Copyright � 1996 by Academic Press, Inc.All rights of reproduction in any form reserved.

* Supported in part by a USA�Israel BSF grant.- E-mail: noga�math.tau.ac.il.� Supported in part by a National Science Foundation postdoctoral fellowship.9 E-mail: ckfan�math.harvard.edu.� E-mail: djk�math.mit.edu.& E-mail: jozsef�math.mit.edu.


This was first conjectured in [1]. The conjecture arises in the study ofthe problem, considered by Wakeford [2], of deciding which sets ofmonomials are removable from a generic homogeneous polynomial using alinear change of variables. For more details, see [1]. The following proofis 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 andorder lexicographically.

Label the set B so that b1<b2<b3< } } } <bm .We first consider the case where d>0 for all d # D.Let f (b1) be the smallest d # D such that b1+d � B. Note that such a d

always exists because m=*[b1+d | d # D]>*[b | b>b1 , b # B]=m&1.Next, let f (b2) be the smallest d # D"[ f (b1)] such that b2+d � B. Such

a d exists for virtually the same reason we are able to define f (b1).Next, let f (b3) be the smallest d # D"[ f (b1), f (b2)] such that b3+d � B.

Again, such a d exists for virtually the same reason we are able to definef (b1) and f (b2).

Continue in this manner until f is defined on all of B.We claim that f is acyclic.To see this, let g be a matching such that mf =mg .If f { g, then there must be a smallest v such that

[b # B | b+ f (b)=v]{[b # B | b+ g(b)=v].

Let b be the smallest element of [b # B | b+g(b)=v] & [b # B | b+f (b)=v]c.Note that f (b)> g(b) since otherwise b+ f (b)<b+ g(b)=v contradicts

our choice of v.On the other hand, if f (b)> g(b), we must have some b$<b for which

f (b$)= g(b), since otherwise f would not have been constructed accordingto our recipe. But since g(b$){ f (b$) (because g(b)= f (b$)), we haveb$+ f (b$)<v again contradicting our choice of v.

This impossibility implies that f = g.For the general case, we partition D into D+ and D& so that D+=

[d # D | d>0]. We now construct a matching f by using the above recipetwice, once for D+ and [b |D&|+1 , ..., bm] and once for D& and [b1 , ..., b |D&|].(For the latter assignment, we use the opposite ordering of Zn.)

We claim that f is acyclic. To see this note that any matching g with mf =mg

must satisfy D&=[ g(b1), ..., g(b |D&|)]. This is because � |D&|k=1 bk+ f (bk) is

an absolute minimum for � |D&|k=1 bk+h(bk) over all matchings h with

equality if and only if D&=[h(b1), ..., h(b |D&|)]. Acyclicity now followsfrom the argument given in the case where all d # D are positive.

235ACYCLIC MATCHINGS


2. FINAL REMARKS

It is worth noting that the number of matchings may be exactly one, forinstance, in the case m=1, or, less trivially, if n=1 and B=D=[1, 2, 3, ..., m].

The result in [1] is not entirely superseded by Theorem 1 since in [1]a connection with hyperplanes is given.

Finally, we remark that throughout we could have replaced Z by Qor R.

REFERENCES

1. C. K. Fan and J. Losonczy, Matchings and canonical forms in symmetric tensors, Adv.Math. 117 (1996), 228�238.

2. E. K. Wakeford, On canonical forms, Proc. London Math. Soc., 2, 18 (1918�1919),403�410.

236 ALON ET AL.

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Acyclic Matchings