Discrete Applied Mathematics 161 (2013) 794–801

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Discrete Applied Mathematics

journal homepage: www.elsevier.com/locate/dam

On the number of perfect matchings of line graphsFengming Dong a, Weigen Yan b,∗, Fuji Zhang c

a National Institute of Education, Nanyang Technological University, Singaporeb School of Sciences, Jimei University, Xiamen 361021, Chinac School of Mathematical Science, Xiamen University, Xiamen 361005, China

a r t i c l e i n f o

Article history:Received 17 February 2011Received in revised form 28 June 2012Accepted 18 October 2012Available online 11 November 2012

Keywords:Perfect matchingLine graphCyclomatic number

a b s t r a c t

Let G = (V , E) be a connected graph, where |E| is even. In this paper we show that the linegraph L(G) of G contains at least 2|E|−|V |+1 perfect matchings, and characterize G such thatL(G) has exactly 2|E|−|V |+1 perfect matchings. As applications, we use a unified approach tosolve the dimer problem on the Kagomé lattice, 3.12.12 lattice, and Sierpinski gasket withdimension two in the context of statistical physics.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

The graphs considered in this paper may have multiple edges but have no loops, if not specified. For a connectedgraph G, let V (G), E(G) and ∆(G) be the vertex set, the edge set and the maximum degree of G respectively. |V (G)| and|E(G)| are called the order (i.e., the number of vertices) and the size (i.e., the number of edges) of G respectively. For anyA = {a1, a2, . . . , as} ⊂ V (G) (resp., E1 = {e1, e2, . . . , et} ⊂ E(G)), let G − A or G − a1 − a2 − · · · − as (resp., G − E1 orG − e1 − e2 − · · · − et ) be the subgraph of G obtained by deleting all vertices in A and all edges incident with vertices in A(resp., by deleting all edges in E1).

For any two edges e1 and e2 in G, let µG(e1, e2) be the number of common ends of e1 and e2 in G. It is clear that0 ≤ µG(e1, e2) ≤ 2. A matching of G is a subset E ′ of E(G) such that µG(e1, e2) = 0 for every pair of edges e1 and e2 inE ′. A perfect matching of G is a matching P such that every vertex in G is incident with some edge in P . Let P (G) be the setof perfect matchings of G and M(G) = |P (G)|. It is well known that computing M(G) of a graph G is an NP-hard problem(see [7,10,19]).

The line graph of a graph G, denoted by L(G), is defined as the graph with V (L(G)) = E(G) such that any two vertices eand f in L(G) are joined by exactly µG(e, f ) edges. It is clear thatM(L(G)) = 0 if |E(G)| is odd. Sumner [16] and Vergnas [20]independently showed that every connected claw-free graph with an even order has a perfect matching. Since every linegraph is claw-free, the line graph of a connected graph with an even size has at least one perfect matching. Obviously, thereexists a one–one correspondence between perfectmatchings of L(G) and decompositions of the edge set of G into pathswithtwo edges (if G is not simple, 2-cycles are allowed as well) (see for example [9,18]).

Perfect matchings of a graph are called dimer configurations in statistical physics [13] and Kekulé structures in quantumchemistry [6,11,17]. These so-called dimer models are the subject of an extensive physics and mathematics literature[2,4,13–15]. By a Pfaffian method, the dimer problem (i.e., the problem of determining the number of perfect matchings ingraphs) on the plane 3.12.12, Kagomé lattices and the Sierpinski gasket has been studied extensively by statistical physicists

∗ Corresponding author. Fax: +86 592 6181893.E-mail addresses: fengming.dong@nie.edu.sg (F. Dong), weigenyan@263.net, weigenyan@jmu.edu.cn (W. Yan), fjzhang@xmu.edu.cn (F. Zhang).

0166-218X/$ – see front matter© 2012 Elsevier B.V. All rights reserved.doi:10.1016/j.dam.2012.10.032

F. Dong et al. / Discrete Applied Mathematics 161 (2013) 794–801 795

(see for example [21–23]). Note that the plane Kagomé, 3.3.12 lattices and the Sierpinski gasket are the line graphs of somegraphs, respectively. It is natural to consider the problem of enumeration of perfect matchings of line graphs.

Let G be a connected graph of order n and size m, where m is even. In Section 2, we find an explicit expression forM(L(G)) when G is a tree. In Section 3, we express M(L(G)) as the summation of M(L(T )) for a set of 2m−n+1 trees T . Thisresult immediately implies that M(L(G)) ≥ 2m−n+1. This section also provides another lower bound for M(L(G)): if G is2-connected, thenM(L(G)) ≥

25 (∆!)1/42m−n+1, where∆ is themaximumdegree ofG. This lower bound is better than 2m−n+1

when ∆ ≥ 5. In Section 4, we characterize all connected graphs G such that M(L(G)) = 2m−n+1. This characterization canbe used to determine the number of perfect matchings of a wide class of graphs which are themselves line graphs of sub-cubic graphs (i.e., graphs of maximum degree at most 3). In particular, in Section 5, we use it to determine the number ofperfectmatchings of the Kagomé lattice, 3.12.12 lattice, and Sierpinski gasketwith dimension two in the context of statisticalphysics.

2. The line graph of a tree

In this section, we will give a formula for M(L(T )) for any tree T of order n. If n is even, then |E(T )| is odd and thusM(L(T )) = 0. Hence, for determiningM(L(T )), we need only to consider the case that n is odd.

For any graph G, let p(G) be the number of those components of G each of which has an even number of edges. If G is aforest, p(G) and |V (G)| have the same parity. Thus, if G is a tree and |V (G)| is odd, then p(G− v) is even for all v ∈ V (G). Forany non-negative integer k, let (2k)!! = (2k)!/(k! × 2k). By induction, it is easy to prove the following result.

Lemma 1. Let T be a tree with V (T ) = {v1, v2, . . . , vn}, where n > 1 is odd. Then

M(L(T )) =

ni=1

p(T − vi)!!.

Two results follow immediately from Lemma 1.

Corollary 1. Let T be a tree of order n,where n > 1 is odd. ThenM(L(T )) ≥ 1, where the equality holds if and only if p(T−v) = 0or 2 for every vertex v in T .

Corollary 2. Let T be a tree with an odd order. If ∆(T ) ≤ 3, then M(L(T )) = 1.

3. Lower bounds forM(L(G))

Let G be a connected graph of order n and size m, where m is even. The cyclomatic number of G, denoted by c(G), isdefined to bem− n+ 1. Thus c(G) ≥ 0, and G is a tree if and only if c(G) = 0. In Section 2, we have obtained an expressionforM(L(T )) for any tree T of odd order. In this section, we shall consider low bounds forM(L(G)) for all connected graphs G.We show thatM(L(G)) ≥ 2c(G) for any connected graph G andM(L(G)) ≥

25 (∆!)1/42c(G) for any 2-connected graph G, where

∆ is the maximum degree of G.We first develop a recursive expression for M(L(G)). Let e be any edge of G with ends u and v. Let G(u, w) be the graph

obtained from G − e by adding a new vertex w and adding a new edge joining w to u. G(v, w) is defined similarly. For anyx ∈ V (G), let Ex be the set of edges in Gwhich are incident with x.

Lemma 2. Let G be a graph and e be an edge of G with ends u and v. Then

M(L(G)) = M(L(G(u, w))) + M(L(G(v, w))).

Proof. Note that a perfectmatching of a line graph corresponds to a decomposition of its edge set into pathswith two edges.In every perfect matching of L(G), e is paired up with an edge e′ which is either in Eu or in Ev . Thus the result follows. �

Let G be a connected graph of order n and size m. Assume that c(G) = m − n + 1 > 0. Then there exists a set E ′ ofm − n + 1 edges e1, e2, . . . , em−n+1 in G such that G − E ′ is a spanning tree of G. Let ui and vi be the two ends of ei fori = 1, 2, . . . ,m − n + 1. For any (m − n + 1)-tuple (j1, j2, . . . , jm−n+1), where ji ∈ {0, 1}, let G(j1, j2, . . . , jm−n+1) be thegraph obtained from G− E ′ by addingm− n+ 1 new vertices w1, w2, . . . , wm−n+1 and for every iwith 1 ≤ i ≤ m− n+ 1,adding a new edge joiningwi to vi if ji = 0 or to ui if ji = 1. Note that G(j1, j2, . . . , jm−n+1) is a tree of orderm+1 and sizem.

Applying Lemma 2 repeatedly yields the following result.

Corollary 3. Let G be a connected graph with n vertices and m edges and m − n + 1 > 0. Then

M(L(G)) =

(j1,j2,...,jm−n+1)

M(L(G(j1, j2, . . . , jm−n+1))), (1)

where the summation ranges over all 2m−n+1 vectors (j1, j2, . . . , jm−n+1), where jk ∈ {0, 1}.

796 F. Dong et al. / Discrete Applied Mathematics 161 (2013) 794–801

By Corollary 3, the number of perfect matchings of the line graph of a connected graph G of order n and size m can beexpressed as the summation of the numbers of perfect matchings of line graphs of 2m−n+1 trees of orderm+1. Note that, byLemma 1, we can easily enumerate perfect matchings of the line graph of a tree. This results in an algorithm to enumerateperfect matchings of the line graph of a connected graph.

We are now going to apply Corollaries 2 and 3 to get a lower bound forM(L(G)).

Theorem 1. Let G be a connected graph with n vertices and m edges, where m is even. Then M(L(G)) ≥ 2m−n+1, where theequality holds if ∆(G) ≤ 3.Proof. By Corollary 3, M(L(G)) is expressed as the sum of 2m−n+1 terms of the form M(L(G(j1, j2, . . . , jm−n+1))), whereG(j1, j2, . . . , jm−n+1) is a tree of orderm + 1 and sizem. ThusM(L(G)) ≥ 2m−n+1 by Corollary 1. If ∆(G) ≤ 3, then each treeG(j1, j2, . . . , jm−n+1) has its maximum degree at most 3, and so M(L(G(j1, j2, . . . , jm−n+1))) = 1 by Corollary 2, implyingthatM(L(G)) = 2m−n+1 in this case. �

Remark 1. Let G be any connected graph with an even number of edges. If there exists a tree G(j1, j2, . . . , jm−n+1) definedin the summation of Corollary 3 and a vertex u in this tree such that G(j1, j2, . . . , jm−n+1) − u has at least three isolatedvertices, thenM(L(G)) > 2m−n+1 by Corollaries 1 and 3.

Remark 2. Little [8] proved that a graph Gwith vertex set V (G) has an even number of perfect matchings if and only if thereis a set S ⊆ V (G), S = ∅, such that every vertex in V (G) is adjacent to an even number of vertices in S. For a connected graphG, if G is not a tree, we may choose S to be the set of edges in a cycle of G, and so in L(G), each vertex in L(G) is adjacent toan even number of vertices in S. By Little’s result,M(L(G)) is even if G is not a tree. We can now verify Little’s result for linegraphs by applying Lemma 1 and Corollary 3. Lemma 1 shows that M(L(G)) is an odd number whenever G is a tree. If G isnot a tree, then c(G) > 0 and by Corollary 3 and Lemma 1,M(L(G)) is the sum of 2c(G) terms each of which is odd, and thusM(L(G)) is even in this case.

Now we are going to find another lower bound for M(L(G)) with respect to its order, size and maximum degree for a2-connected graph G.

For any integer r ≥ 0, define

η(r) =

0≤k≤r/2

r2k

(2k)!!.

Lemma 3. Let G be a connected graph with n vertices and m edges, where m is even. Let x be any vertex in G such that G − x isconnected. Then

M(L(G)) ≥ η(d(x)) · 2m−n−d(x)+2.

Proof. For any subset S ⊆ Ex, we define a new graph, denoted by GS . If S = {xu1, xu2, . . . , xus} ⊆ Ex, let GS be the graphobtained from G−x by adding s new verticesw1, w2, . . . , ws and s new edges joining ui towi for all i = 1, 2, . . . , s. ApplyingLemma 2 repeatedly yields that

M(L(G)) =

S⊆Ex

M(L(GS))M(L(T1,d(x)−|S|)),

where T1,t denotes the star with t + 1 vertices. Note thatM(L(T1,t)) = t!! if t is even andM(L(T1,t)) = 0 otherwise. Thus

M(L(G)) =

0≤k≤d(x)/2

(2k)!!S⊆Ex

|S|=d(x)−2k

M(L(GS)).

As GS is a connected graphwith n−1+|S| vertices andm−d(x)+|S| edges, we haveM(L(GS)) ≥ 2m−n−d(x)+2 by Theorem 1.Thus

M(L(G)) ≥

0≤k≤d(x)/2

(2k)!!S⊆Ex

|S|=d(x)−2k

2m−n−d(x)+2

= 2m−n−d(x)+2

0≤k≤d(x)/2

(2k)!!d(x)2k

= η(d(x))2m−n−d(x)+2. �

The following result gives lower bounds for η(r).

Lemma 4. For any integer r ≥ 0, η(r) ≥2

√30

√(r + 2)! and η(r) ≥

2r5 (r!)1/4.

Proof. Note that η(0) = η(1) = 1. It can be shown that η(r) = η(r − 1) + (r − 1)η(r − 2) for r ≥ 2. It is not difficultto verify that η(r) ≥

2√30

√(r + 2)! holds for r ≤ 10. For r > 10, this inequality can be proved inductively by applying the

recursive formula η(r) = η(r − 1) + (r − 1)η(r − 2). The second inequality can be proved similarly. �

F. Dong et al. / Discrete Applied Mathematics 161 (2013) 794–801 797

By Lemmas 3 and 4, we obtain the following lower bound forM(L(G)) for 2-connected graph G.

Theorem 2. Let G be a 2-connected graph with n vertices, m edges and maximum degree ∆, where m is even. Then

M(L(G)) ≥2

√30

((∆ + 2)!)1/22m−n−∆+2 and M(L(G)) ≥25(∆!)1/42m−n+1.

Note that the first bound in the above result is better than the second one if and only if ∆ ≤ 7 or ∆ ≥ 16, and the secondbound is better than the bound 2m−n+1 if and only if ∆ ≥ 5. However, both bounds in Theorem 2 are much better than2m−n+1 when ∆ is large.

4. The extremal graphs G withM(L(G)) = 2m−n+1

In this section, we characterize those connected graphs G such that the equalityM(L(G)) = 2m−n+1 holds.We first obtainthe following result by Corollary 3 and Remark 1.

Corollary 4. Let G be a connected graph with n vertices and m edges, where m is even. If ∆(B) ≥ 4 for some block B of G, thenM(L(G)) > 2m−n+1.

Proof. Note that M(L(G)) is expressed as the summation in Corollary 3. If ∆(B) ≥ 4 for some block B of G, then there is asubset E ′ of E(G) with |E ′

| = m − n + 1 such that G − E ′ is a spanning tree of G and at least three edges of E ′ are incidentwith a vertex in B. Thus some tree G(j1, j2, . . . , jm−n+1) in the summation of Corollary 3 has a vertex which is adjacent to atleast 3 vertices of degree 1, implying thatM(L(G)) > 2m−n+1 by Remark 1. �

Theorem 1 and Corollary 4 immediately yield the following result.

Corollary 5. Let G be a 2-connected graph of order n and size m, where m is even. Then M(L(G)) ≥ 2m−n+1, where the equalityholds if and only if ∆(G) ≤ 3.

If G is not 2-connected, the condition that ∆(B) ≤ 3 holds for every block B of G is not enough to guarantee thatM(L(G)) = 2m−n+1. For example, if G is a tree, it is possible that M(L(G)) > 2m−n+1. In the following, we shall considerthe case that G is not 2-connected and find a necessary and sufficient condition for the equalityM(L(G)) = 2m−n+1.

For any connected graph G and any u ∈ V (G), let gG(u) = p(G − u) and fG(u) = dG(u) − k, where dG(u) is the degree ofu in G and k is the number of components of G − u. Note that fG(u) ≥ 0, where the equality holds if and only if every edgeof Eu is a bridge of G.

The next result follows immediately.

Lemma 5. If u is a cut-vertex of G, and G1 and G2 are two subgraphs of G such that V (G1)∩V (G2) = {u} and E(G1)∪ E(G2) =

E(G), then

fG(u) = fG1(u) + fG2(u) and gG(u) = gG1(u) + gG2(u).

Lemma 6. Let G be any connected graph with m edges. For any vertex u in G,

fG(u) + gG(u) ≡ m(mod 2).

Proof. It is easy to verify if u is not a cut-vertex, then fG(u) + gG(u) is even if and only if |E(G)| is even. If u is a cut-vertex,by Lemma 5,

fG(u) + gG(u) =

ki=1

(fGi(u) + gGi(u)),

where Gi is the subgraph of G induced by V (G′

i) ∪ {u} for i = 1, 2, . . . , k, and G′

1,G′

2, . . . ,G′

k are the components of G − u.Thus

fG(u) + gG(u) =

ki=1

(fGi(u) + gGi(u)) ≡

ki=1

|E(Gi)| = m(mod 2). �

Lemma 7. Let G be any connected graph and e be an edge with ends u and v. If e is on some cycle of G, then for every x ∈ V (G),

0 ≤ (fG(x) + gG(x)) − (fG(u,w)(x) + gG(u,w)(x)) ≤ 2.

Proof. Since e is an edge on some cycles ofG,G−e is also connected. It is easy to verify that fG(u,w)(u) = fG(u)−1, gG(u,w)(u) =

gG(u) + 1, fG(u,w)(v) = fG(v) − 1 and gG(v) − 1 ≤ gG(u,w)(v) ≤ gG(v) + 1, implying that the inequality of the lemma holdsfor x ∈ {u, v}.

798 F. Dong et al. / Discrete Applied Mathematics 161 (2013) 794–801

Now consider the case that x ∈ V (G) \ {u, v}. Observe that e is in some component of G − x, say H . If e is nota bridge of H , then fG(u,w)(x) = fG(x) and gG(u,w)(x) = gG(x). If e is a bridge of H , then fG(u,w)(x) = fG(x) − 1 andgG(x) − 1 ≤ gG(u,w)(x) ≤ gG(x) + 1. Thus the inequality of the lemma also holds in this case. �

We are now in a position to characterize all those connected graphs G with M(L(G)) = 2m−n+1, where n = |V (G)| andm = |E(G)|.

Theorem 3. Let G be a connected graph with n vertices and m edges, where m is even. Then the following statements areequivalent:

1. M(L(G)) = 2m−n+1;2. fG(u) + gG(u) ∈ {0, 2} holds for every vertex u in G;3. ∆(B) ≤ 3 for every block B of G and fG(u) + gG(u) ∈ {0, 2} for every cut-vertex u in G.

Proof. Note that by Lemma 6, fG(u) + gG(u) ∈ {0, 2} if and only if fG(u) + gG(u) ≤ 2.(i) ⇒ (ii) Suppose that fG(u) + gG(u) ≥ 3 for some vertex u in G. We shall show that (i) does not hold. We first consider

the case that fG(u) = 0. In this case, every edge of Eu is a bridge of G. Let {e1, e2, . . . , em−n+1} be a set of edges in G such thatG − {e1, e2, . . . , em−n+1} is a spanning tree of G. So ei ∈ Eu for all i = 1, 2, . . . ,m − n + 1.

Note that G(j1, j2, . . . , jm−n+1) − u has exactly k components, each of which is also a tree, where k is the number ofcomponents of G − u.

Claim A. For any component F of G(j1, j2, . . . , jm−n+1) − u and any component H of G − u, either V (F) ∩ V (H) = ∅ orV (H) ⊆ V (F); and further, if V (H) ⊆ V (F), then |V (F)| = |E(H)| + 1.

It is clear that either V (F) ∩ V (H) = ∅ or V (H) ⊆ V (F). Assume that V (H) ⊆ V (F). By the definition ofG(j1, j2, . . . , jm−n+1), we have |E(F)| = |E(H)|. As F is a tree, |V (F)| = |E(F)| + 1 = |E(H)| + 1. The claim holds.

By Claim A, G(j1, j2, . . . , jm−n+1) − u has exactly gG(u) components each of which has an odd number of vertices. Sop(G(j1, j2, . . . , jm−n+1) − u) = gG(u) ≥ 3, implying that M(L(G(j1, j2, . . . , jm−n+1))) ≥ 3 by Corollary 1. Then Corollary 3implies thatM(L(G)) > 2m−n+1.

Assume that M(L(G)) > 2m−n+1 when 0 ≤ fG(u) < t , where t ≥ 1. Now consider the case that fG(u) = t . As fG(u) > 0,there exists an edge e ∈ Eu such that e is not a bridge of G. Let e = uv. Note that fG(u,w)(u) = fG(u) − 1 and gG(u,w)(u) =

gG(u) + 1, implying that fG(u,w)(u) + gG(u,w)(u) ≥ 3, where G(u, w) is defined in Lemma 2. As fG(u,w)(u) < fG(u) = t ,by induction, we have M(L(G(u, w))) > 2m−(n+1)+1

= 2m−n. By Theorem 1, M(L(G(v, w))) ≥ 2m−(n+1)+1= 2m−n. Then

Lemma 2 implies that

M(L(G)) = M(L(G(u, w))) + M(L(G(v, w))) > 2m−n+1.

Hence (ii) follows from (i).(ii) ⇒ (i). Suppose that M((L(G))) > 2m−n+1. We may assume that G has the minimum cyclomatic number c(G) among

all such graphs.Suppose that c(G) = 0. Then G is a tree, implying that fG(u) = 0 and p(G − u) = gG(u) = fG(u) + gG(u) ≤ 2 for every

u ∈ V (G). By Corollary 1, we haveM(L(G)) = 1 = 2m−n+1, contradicting the assumption on G.ThusG contains cycles. Let e be an edge in some cycle ofG and u and v be its ends. BothG(u, w) andG(v, w) are connected

graphs of order n + 1 and size m. By Lemma 7, condition (ii) holds for G(u, w). Since G(u, w) has a smaller cyclomaticnumber than G, by the assumption of G, we have M((L(G(u, w)))) = 2m−(n+1)−1

= 2m−n. Similarly, it is also true thatM((L(G(v, w)))) = 2m−n. Thus Lemma 2 implies that M(L(G)) = 2m−n+1, a contradiction. Hence (i) follows from (ii).

(ii) ⇒ (iii). Let B be any block of G. If dB(u) ≥ 4, then fG(u) ≥ 3, contradicting the condition that f (u) + g(u) ∈ {0, 2}.(iii) ⇒ (ii). Let u be a vertex in G which is not a cut-vertex of G. So u is contained in only one block of G, say B. Then

dG(u) = dB(u) ≤ ∆(B) ≤ 3. So fG(u) + gG(u) ≤ 2 + 1. Since m is even, fG(u) + gG(u) is even by Lemma 6, and thereforefG(u) + gG(u) ∈ {0, 2}. �

5. Applications of Theorem 1

Recall that Theorem 1 provides an exact formula for the number of perfect matchings of those graphs which arethemselves line graphs of sub-cubic graphs. In this sectionweapply this result to enumerate perfectmatchings of a Sierpinskigasket with dimension two, 3.12.12 lattice, and Kagomé lattice in the context of statistical physics.

5.1. The two-dimensional Sierpinski gasket

The construction of the two-dimensional Sierpinski gasket, denoted by SG2(n) at stage n is shown in Fig. 1. At stage n = 0,it is an equilateral triangle; while stage n + 1 is obtained by the juxtaposition of three n-stage structures. It is not difficultto see that SG2(n) has 3

2 (3n+ 1) vertices and 3n+1 edges. The dimer problem on the two-dimensional Sierpinski gasket was

solved by Chang and Chen [1], who proved the following result.

F. Dong et al. / Discrete Applied Mathematics 161 (2013) 794–801 799

Fig. 1. The first four stages n = 0, 1, 2, 3 of two-dimensional Sierpinski gasket SG2(n).

Fig. 2. The first four stages n = 0, 1, 2, 3 of the graphs Gn .

Theorem 4 (Chang and Chen, [1]). Suppose SG2(n) is the two-dimensional Sierpinski gasket. Then the number of perfectmatchings of SG2(n) equals 2(3n−1)/2 if n is odd.

We shall give a new proof of Theorem 4 by Theorem 1. For each n ≥ 0, we construct a graph Gn such that L(Gn) isisomorphic to SG2(n). In fact, G0 is the star K1,3 and Gn+1 is obtained by the juxtaposition of three Gn, as shown in Fig. 2. It iseasily verified that L(Gn) ∼= SG2(n).

By the definition of Gn,Gn has order 3n+ 3 and size 3

2 (3n+ 1). Obviously, Gn has an even size if and only if n is odd. Since

∆(Gn) = 3, if n is odd, by Theorem 1,

M(SG2(n)) = M(L(Gn)) = 232 (3n+1)−(3n+3)+1

= 2(3n−1)/2.

This is another proof of Theorem 4.

5.2. 3.12.12 lattice

The 3.12.12 lattice RT (n,m) with toroidal boundary condition is shown in Fig. 3(a), where (ai, a∗

i ) and (bj, b∗

j ) are edgesin RT (n,m) for all i = 1, 2, . . . ,m + 1 and j = 1, 2, . . . , n + 1. The 3.12.12 lattice RT (n,m) was used by Fisher [5] in adimer formulation of the Isingmodel. Bymeans of Pfaffians, Fisher [5] andWu [22] proved that the logarithm of the numberof perfect matchings of RT (n,m), divided by 3(m + 1)(n + 1) (the number of edges of each of the perfect matchings ofRT (n,m)), converges to 1

3 ln 2 asm, n → ∞, which is called the entropy of RT (n,m) by statistical physicists. We shall deriveby Theorem 1 a formula for the numbers of perfect matchings of RT (n,m) as shown below.

Theorem 5. Let RT (n,m) be the 3.12.12 lattice with toroidal boundary condition. Then

M(RT (n,m)) = 2mn+m+n+2.

Proof. In order to prove the theorem, we introduce the hexagonal lattices which have been extensively studied bystatistical physicists [3,5,22]. The hexagonal lattice HT (n,m) with toroidal boundary condition is shown in Fig. 3(b), where(d1, d∗

1), (d2, d∗

2), . . . , (dm+1, d∗

m+1) and (d1, c∗

1 ), (c1, c∗

2 ), . . . , (cn−1, c∗n ), (cn, d

∗

m+1) are edges in HT (n,m). It is not difficultto see that the line graph of S(HT (n,m)) is RT (n,m), where S(HT (n,m)) is the graph obtained from HT (n,m) by subdividingeach edge of HT (n,m) once. Note that S(HT (n,m)) has order 5(m+1)(n+1), size 6(m+1)(n+1) andmaximum degree 3.By Theorem 1, we have

M(RT (n,m)) = M(L(S(HT (n,m)))) = 26(m+1)(n+1)−5(m+1)(n+1)+1= 2mn+m+n+2. �

800 F. Dong et al. / Discrete Applied Mathematics 161 (2013) 794–801

a b

Fig. 3. (a) The 3.12.12 lattice RT (n,m) with toroidal boundary condition. (b) The hexagonal lattice HT (n,m).

Fig. 4. The graph G(n,m).

5.3. Kagomé lattices

Let G(n,m) be the plane lattice graph illustrated with solid lines in Fig. 4, each of whose vertices has degree two or four.For G(n,m), if we identify each pair of vertices ui and u∗

i , for all i = 1, 2, . . . , 2m, and vj and v∗

j for all j = 1, 2, . . . , n, theresulting graph, denoted by K T (n,m), is called the Kagomé lattice with toroidal boundary condition by statistical physicists(see [3,12,21–23]).

The study of the molecular freedom for the Kagomé lattice has been a subject matter of interest for many years (see,for example, [3,12]), but most of the studies have been numerical or approximate. By using Pfaffian orientation, Wu andWang [23] obtained a formula for the numbers of perfect matchings of K T (n,m): M(K T (n,m)) = 22mn+1. Now we give anew proof for this formula.

By the definition of K T (n,m), we know that K T (n,m) is actually the line graph of HT (n − 1, 2m − 1) in Fig. 3(b), whichis also illustrated by dashed lines in Fig. 4. Note that HT (n − 1, 2m − 1) has order 4mn, size 6mn and maximum degree 3.Thus, by Theorem 1, we have

M(K T (n,m)) = M(L(HT (n − 1, 2m − 1))) = 26mn−4mn+1= 22mn+1.

Acknowledgments

This work was partially finished while the second author visited the National Institute of Education, Singapore. We alsothank the referees for some valuable suggestions and help in great detail to make this paper to be more pleasant to read.One of the referees told us that there exists a correspondence between perfect matchings of L(G) and decompositions of theedge set of G into paths with two edges, which may simplify the proof of some of our results.

The first author was supported by NIE AcRf funding (RI 5/06 DFM) of Singapore. The second author was supported by theNSFC Grant (11171134) and the Foundation for Young Professors of Jimei University. The third author was supported by theNSFC Grant (10831001).

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