On the Number of Matchings in Regular Graphs

# On the Number of Matchings in Regular Graphs

S. Friedland, E. Krop and K. Markström Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60607-7045, USA (friedlan@uic.edu).Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60607-7045, USA (ekrop1@math.uic.edu).Department of Mathematics and Mathematical Statistics, UmeåUniversity, SE-901 87 Umeå, Sweden
12 January, 2008
###### Abstract

For the set of graphs with a given degree sequence, consisting of any number of and , and its subset of bipartite graphs, we characterize the optimal graphs who maximize and minimize the number of -matchings.

We find the expected value of the number of -matchings of -regular bipartite graphs on vertices with respect to the two standard measures. We state and discuss the conjectured upper and lower bounds for -matchings in -regular bipartite graphs on vertices, and their asymptotic versions for infinite -regular bipartite graphs. We prove these conjectures for -regular bipartite graphs and for -matchings with .
2000 Mathematics Subject Classification: 05A15, 05A16, 05C70, 05C80, 82B20
Keywords and phrases: Partial matching and asymptotic growth of average matchings for -regular bipartite graphs, asymptotic matching conjectures.

## 1 Introduction

Let be an undirected graph with the set of vertices and the set of edges . An -matching , is a set of distinct edges in , such that no two edges have a common vertex. We say that covers , if the set of vertices incident to is . Denote by the number of -matchings in . If is even then -matching is called a perfect matching, or -factor of , and is the number of -factors in . For an infinite graph , a match is a match of density , if the proportion of vertices in covered by is . Then the -matching entropy of is defined as

 hG(p)=limsupk→∞logϕ(mk,Gk)#Vk,

where is a sequence of finite graphs converging to , and . See for details .

The object of this paper is two folds. First we consider the family , the set of simple graphs on vertices with vertices of degree and vertices of degree . Let be the subset of bipartite graphs. For each we characterize the optimal graphs which maximize and minimize for and . It turns out the optimal graphs do not depend on but on and . Furthermore, the graphs with the maximal number of -matchings, are bipartite.

Second, we consider , the set of simple bipartite -regular graphs on vertices, where . Denote by a cycle of length and by the complete bipartite graph with -vertices in each group. For a nonnegative integer and a graph denote by the disjoint union of copies of . Let

 λ(m,n,r):=minG∈G(2n,r)ϕ(m,G),Λ(m,n,r):=maxG∈G(2n,r)ϕ(m,G), m=1,…,n. (1.1)

Our results on -regular graphs yield.

 λ(m,n,2)=ϕ(m,C2n), (1.2) Λ(m,2q,2)=ϕ(m,qK2,2),Λ(m,2q+3,2)=ϕ(m,qK2,2∪C6), (1.3) for m=1,…,n.

The equality inspired us to conjecture the Upper Matching Conjecture, abbreviated here as UMC:

 Λ(m,qr,r))=ϕ(m,qKr,r) for m=1,…,qr. (1.4)

For the value the UMC follows from Bregman’s inequality . For the value the UMC holds up to . The results of  support the validity of the above conjecture for and large values of . As in the case we conjecture that that for any nonbipartite -regular graph on vertices for .

It is useful to consider , the set of -regular bipartite graphs on vertices, where multiple edges are allowed. Observe that , where is the -regular multi-bipartite graph on vertices. Let

 μ(m,n,r):=minG∈Gmult(2n,r)ϕ(m,G),M(m,n,r):=maxG∈Gmult(2n,r)ϕ(m,G), (1.5) m=1,…,n,2≤r∈N.

It is straightforward to show that

 M(m,n,r)=ϕ(m,nHr)=(nm)rm,m=1,…,n. (1.6)

Hence for most of the values of . On the other hand, as in the case of , it is plausible to conjecture that for all allowable values and .

It was shown by Schrijver  that for

 ϕ(n,G)≥((r−1)r−1rr−2)n,forallG∈Gmult(2n,r). (1.7)

This lower bound is asymptotically sharp. In the first version of this paper we stated the conjectured lower bound

 ϕ(m,G)≥(nm)2(nr−mnr)rn−m(mrn)m, (1.8) forallG∈Gmult(2n,r)(2n,r) and m=1,…,n.

Note that for the above inequality reduces to (1.7). Our computations suggest a slightly stronger version of the above conjecture (7.1).

Recently Gurvits  improved (1.7) to

 ϕ(n,G)≥r!rr(rr−1)r(r−1)((r−1)r−1rr−2)n,G∈Gmult(2n,r). (1.9)

In  the authors were able to generalize the above inequality to partial matching, which are very close to optimal results asymptotically, see  and below.

The next question we address is the expected value of the number of -matchings in . There are two natural measures on , [7, Ch.9] and [8, Ch.8]. Let be the expected value of with respect to the measure for . In this paper we show that

 limk→∞logEi(mk,nk,r)2nk=ghr(p), for i=1,2, (1.10) iflimk→∞nk=limk→∞mk=∞,andlimk→∞mknk=p∈[0,1], (1.11) ghr(p):=12(plogr−plogp−2(1−p)log(1−p)+(r−p)log(1−pr)). (1.12)

In view of (1.10) the inequalities (1.7) and (1.9) give the best possible exponential term in the asymptotic growth with respect to , as stated in . Similarly, the conjectured inequality (1.8), if true, gives the best possible exponential term in the asymptotic growth with respect to , and .

For let be the infimum of over all sequences satisfying (1.11). Hence for any infinite bipartite -regular graph. Clearly . We conjecture

 lowr(p)=ghr(p). (1.13)

(1.2) implies the validity of this conjecture for . The results of  imply the validity of this conjecture for each and any . In  we give lower bounds on for each and which are very close to .

We stated first our conjectures in the first version of this paper in Spring 2005. Since then the conjectured were restated in [3, 4] and some progress was made toward validations of these conjectures.

We now survey briefly the contents of this paper. In §2 we give sharp bounds for the number of -matchings for general and bipartite -regular graphs. In §3 we generalize these results to . In §4 we find the average of -matchings in -regular bipartite graphs with respect to the two standard measures. We also show the equality (1.10). In §5 we discuss the Asymptotic Lower Matching Conjecture. In §6 we discuss briefly upper bounds for matchings in -regular bipartite graphs. In §7 we bring computational results for regular bipartite graphs on at most vertices. We verified for many of these graphs the LMC and UMC. Among the cubic bipartite graphs on at most vertices we characterized the graphs with the maximal number of -matching in the case is not divisible by . In §8 we find closed formulas for for and any . It turns out that and depend only on and . , where is the number of cycles in . and equality holds if and only if .

## 2 Sharp bounds for matching of 2-regular graphs

In this section we find the maximal and the minimal matching of -regular bipartite and non-bipartite graphs on vertices. First we introduce the following partial order on the algebra of polynomials with real coefficients, denoted by . By we denote the zero polynomial.

For any two polynomials we let , or , if and only if all the coefficients of are nonnegative. We let if and . Let be the cone of all polynomial with nonnegative coefficients in . Then . Furthermore, if then unless and .

Denote . Let be a graph on vertices. We will identify with . We agree that . Denote by the generating matching polynomial

 ΦG(x):=⌊n2⌋∑m=0ϕ(m,G)xm=∞∑m=0ϕ(m,G)xm. (2.1)

It is straightforward to show that for any two graphs we have the equality

 ΦG∪G′(x)=ΦG(x)ΦG′(x). (2.2)

Denote by a path on vertices: . View each match as an edge. Then an -matching of is composed of edges and vertices. Altogether objects. Hence the number of -matchings is equal to the number of different ways to arrange edges and vertices on a line. Thus

 ϕ(Pk,m)=(k−mm)for m=1,…,⌊k2⌋, (2.3) pk(x):=ΦPk(x)=⌊k2⌋∑m=0(k−mm)xm=∞∑m=0(k−mm)xm. (2.4)

It is straightforward to see that satisfy the recursive relation

 pk(x)=pk−1(x)+xpk−2(x),k=2,…, (2.5) where p1(x)=1,ΦP0(x):=p0(x)=1.

Indeed, . Assume that . All matchings of , where the vertex is not in the matching, generate the polynomial . All matchings of , where the vertex is in the matching, generate the polynomial . Hence the above equality holds. Observe next

 qk(x):=ΦCk(x)=pk(x)+xpk−2(x),k=3,… (2.6)

Indeed, is the contribution from all matching which does not include the matching . The polynomial corresponds to all matchings which include the matching .

Use (2.5) to deduce

 qk(x)=qk−1(x)+xqk−2(x),k=3,…, (2.7) where ΦC2:=q2(x)=1+2x,ΦC1:=q1(x)=1.

Note that we identify with the -regular multibipartite graph . It is useful to consider (2.5) for and (2.6) for . This yields the equalities:

 ΦP−1(x)=p−1=0,ΦP−2(x)=p−2=1x,ΦC0(x)=q0=2. (2.8)

Clearly

 p−1=0≺p0=p1=q1=1≺q0=2,p2=1+x≺q2=p3=1+2x, (2.9) pn≺qn≺pn+1 for all integers n≥3. (2.10)
###### Theorem 2.1

Let be nonnegative integers. Then

 ΦCi(x)ΦCj(x)−ΦCi+j(x)=(−1)ixiΦCj−i(x). (2.11)

In particular, if is even, and if is odd.

Proof. We use the notation for . The case follows immediately from . The case follows from and the identity (2.6) for : . We prove the other cases of the theorem by induction on . Assume that the theorem holds for , where . Let . Then for use (2.6) for and the induction hypothesis for and to obtain:

 ql+1qj−ql+1+j=(ql+xql−1)qj−(ql+j+xql−1+j)= qlqj−ql+j+x(ql−1qj−ql−1+j)=(−1)l+1xl(−qj−l+qj−l+1)=(−1)l+1xl+1qj−l−1.

Hence (2.11) holds. Since for (2.11) implies the second part of the theorem.

###### Theorem 2.2

Let be a -regular graph on vertices. Then

 ΦG(x)⪯ΦC4(x)n4if 4|n (2.12) ΦG(x)⪯ΦC4(x)n−54ΦC5(x)if 4|n−1, (2.13) ΦG(x)⪯ΦC4(x)n−64ΦC6(x)if 4|n−2, (2.14) ΦG(x)⪯ΦC4(x)n−74ΦC7(x)if 4|n−3, (2.15) ΦG(x)⪰ΦC3(x)n3if 3|n (2.16) ΦG(x)⪰ΦC3(x)n−43ΦC4(x)if 3|n−1, (2.17) ΦG(x)⪰ΦC3(x)n−53ΦC5(x)if 3|n−2. (2.18)

Equalities in (2.12-2.15) hold if and only if is either a union of copies of , or a union of copies of and a copy of for , respectively. Equalities in (2.16-2.18) hold if and only if is either a union of copies of , or a union of copies of and a copy of for , respectively.

Assume that is even and is a multi-bipartite -regular graph. Then . Equality holds if and only if .

Proof. Recall that any -regular graph is a union of cycles of order at least. Use (2.2) to deduce that the matching polynomial of is the product of the matching polynomials of the corresponding cycles.

We discuss first the upper bounds on . If and are two odd cycle Theorem 2.1 yields that , where is an even cycle. To find the upper bound on we may assume that contains at most one odd cycle. For all cycles , where Theorem 2.1 yields the inequality . Use repeatedly this inequality, until we replaced the products of different with products involving , and perhaps one factor of the form where . Use (2.11) to obtain the inequality:

 q34=q4(q8+2x4)=q12+3x4q4≻q12+2x6=q26.

Hence we may assume that contains at most one cycle of length . If is even we deduce that we do not have a factor corresponding to an odd cycle, and we obtain the inequalities (2.12) and (2.14). Assume that is odd. Use (2.11) to deduce

 q3q4≺q7,q3q6≺q9≺q4q5,q5q6≺q11≺q4q7, q24q5=q4(q9+x4)=q13+x4q5+x4q4≻q13+x6=q6q7.

These inequalities yield (2.13) and (2.15). Equality in (2.12-2.15) if and only if we did not apply Theorem 2.1 at all.

We discuss second the lower bounds on . If then we use the inequality . Use repeatedly this inequality, until we replaced the products of different with products involving , and . As

 q24≻q8≻q3q5,q4q5≻q9≻q33,q25=q10−2x5=q3q7+x3q4−2x5≻q3q7≻q23q4,

we deduce (2.16-2.18). Equalities hold if we did not apply Theorem 2.1 at all.

Assume finally that is a -multi regular bipartite graph on vertices. Then is a union of even cycles for . Assume that and are even cycles. Then Theorem 2.1 yields that . Continue this process until we deduce that . Equality holds if and only if .

Use Theorem 2.2 and Theorem 2.1 for to deduce.

###### Corollary 2.3
• Let be a simple -regular graph on vertices. Then . Equality holds if and only if .

• Let be a -multi regular graph on vertices. Then . Equality holds if and only if .

Note that the above results verify all the claims we stated about -regular bipartite graphs in the Introduction.

## 3 Graphs of degree at most 2

Denote by the set of simple graphs and multigraphs on vertices respectively, which have vertices of degree and the rest vertices have degree degree . The following proposition is straightforward.

###### Proposition 3.1
• Each is a union of paths and possibly cycles for .

• Each is a union of paths and possibly cycles for .

if and only if .

Denote by the subset of graphs on vertices which are union of -paths. Note that . As in §2 we study the minimum and maximum m-matchings in .

We first study the case where , i.e. is a union of two paths with the total number of vertices equal to .

###### Lemma 3.2

Let . Then

• If mod then

 pn−1=p1pn−1≺p3pn−3≺⋯≺p2⌊n4⌋−1pn−2⌊n4⌋+1≺ (3.1) p2⌊n4⌋pn−2⌊n4⌋≺p2⌊n4⌋−2pn−2⌊n4⌋+2≺⋯≺p2pn−2≺p0pn=pn.
• If mod then

 pn−1=p1pn−1≺p3pn−3≺⋯≺p2⌊n4⌋+1pn−2⌊n4⌋−1≺ (3.2) p2⌊n4⌋pn−2⌊n4⌋≺p2⌊n4⌋−2pn−2⌊n4⌋+2≺⋯≺p2pn−2≺p0pn=pn.

Proof.  Let and consider the path . By considering the generating matching polynomial without the match and with match we get the identity

 pi+j=pipj+xpi−1pj−1 (3.3)

Hence . Subtracting from this equation (3.3) we obtain . Assume that . Continuing this process times, and taking in account that we get

 pi−1pj+1−pipj=(−1)i−1xipj−i for 0≤i≤j+2. (3.4)

Hence . Add this equation to the previous one and use (2.5) to obtain

 pi−2pj+2−pipj=(−1)i−2xi−1pj−i+1 for 1≤i≤j+2. (3.5)

We now prove (3.1-3.2). In (3.5) assume that is odd and . So . Hence . This explains the ordering of the polynomials appearing in the first line of (3.1-3.2). Assume now that is even and . So . Hence . This explains the ordering of the polynomials appearing in the second line line of (3.1-3.2).

The last inequality in the first line of (3.1-3.2) is implied by (3.4).

###### Theorem 3.3

Let . Then for any

 ΦJ⪯ΦG⪯ΦK. (3.6)

Equality in the left-hand side and right-hand side holds if and only if and respectively. Here and is defined as follows:

1. If then .

2. If then .

Proof.  For the theorem follows from Lemma 3.2. For apply the theorem for for any two paths in to deduce that and are the maximal and the minimal graphs respectively.

We extend the result of Lemma 3.2 for cycles.

###### Lemma 3.4

Let . Then

• If mod then

 qn−1=q1qn−1≺q3qn−3≺⋯≺q2⌊n4⌋−1qn−2⌊n4⌋+1≺ (3.7) q2⌊n4⌋qn−2⌊n4⌋≺q2⌊n4⌋−2qn−2⌊n4⌋+2≺⋯≺q2qn−2≺qn+1.
• If mod then

 qn−1=q1qn−1≺q3qn−3≺⋯≺q2⌊n4⌋+1qn−2⌊n4⌋−1≺ (3.8) q2⌊n4⌋qn−2⌊n4⌋≺q2⌊n4⌋−2qn−2⌊n4⌋+2≺⋯≺q2qn−2≺qn+1.

Proof.  The equality (2.7) implies

 qn+1=qn+xqn−1=qn−1+xqn−2+xqn−2+x2qn−3≻qn−2+2xqn−2=q2qn−2.

Hence the last inequality in (3.7) and (3.8) holds. By (2.11) we have . Using this, it is easy to see that

 qi−1qj+1−qiqj=(−1)i−1xi−1qj−i+2−(−1)ixiqj−i=(−1)i−1xi−1(qj−i+2+xqj−i),

as well as

 qi−2qj+2−qiqj=(−1)i−2xi−2qj−i+4−(−1)ixiqj−i=(−1)i−2xi−2(qj−i+4−x2qj−i)= (−1)i−2xi−2(qj−i+3+xqj−i+2−x2qj−i)=(−1)i−2xi−2(qj−i+3+xqj−i+1).

Compare these equalities with (3.4) and (3.5) we obtain all other inequalities in (3.7) and (3.8).

Next, we study graphs composed of a path and a cycle of the form .

###### Lemma 3.5

Let . Then

• If mod then

 qn−1=p1qn−1≺q3pn−3≺p3qn−3≺q5pn−5≺p5qn−5≺⋯≺q2⌊n4⌋−1pn−2⌊n4⌋+1≺ p2⌊n4⌋−1qn−2⌊n4⌋+1≺p2⌊n4⌋qn−2⌊n4⌋⪯q2⌊n4⌋pn−2⌊n4⌋≺p2⌊n4⌋−2qn−2⌊n4⌋+2≺q2⌊n4⌋−2pn−2⌊n4⌋+2 ≺⋯≺p4qn−4≺q4pn−4≺p2qn−2≺q2pn−2≺p0qn=qn. (3.9)

(If mod then is , and otherwise is .)

• If mod then

 qn−1=p1qn−1≺q3pn−3≺p3qn−3≺⋯≺q2⌊n4⌋+1pn−2⌊n4⌋−1⪯p2⌊n4⌋+1qn−2⌊n4⌋−1≺ p2⌊n4⌋qn−2⌊n4⌋≺q2⌊n4⌋pn−2⌊n4⌋≺p2⌊n4⌋−2qn−2⌊n4⌋+2≺q2⌊n4⌋−2pn−2⌊n4⌋+2≺ ⋯≺p4qn−4≺q4pn−4≺p2qn−2≺q2pn−2≺p0qn=qn. (3.10)

(If mod then is , and otherwise is .)

Proof. Assume that . Use (2.6) to obtain

 piqj−qi+2pj−2=pi(pj+xpj−2)−(pi+2+xpi)pj−2=pipj−pi+2pj−2.

(3.5) implies

 piqj−qi+2pj−2=(−1)ixi+1pj−i−3 if i≤j−3, (3.11) piqj−qi+2pj−2=(−1)j−1xj−1pi−j+1 if % i≥j−2 (3.12)

Assume that . Hence, if is odd we get that . If is even then . These inequalities yield slightly less than the half of the inequalities in (3.9) and (3.10).

Assume that . Use (2.6) and (