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On the number of perfect matchings of line graphs
Discrete Applied Mathematics 161 (2013) 794–801
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On the number of perfect matchings of line graphs Fengming Dong a , Weigen Yan b,∗ , Fuji Zhang c a
National Institute of Education, Nanyang Technological University, Singapore
b
School of Sciences, Jimei University, Xiamen 361021, China
c
School of Mathematical Science, Xiamen University, Xiamen 361005, China
article
info
Article history: Received 17 February 2011 Received in revised form 28 June 2012 Accepted 18 October 2012 Available online 11 November 2012 Keywords: Perfect matching Line graph Cyclomatic number
abstract Let G = (V , E ) be a connected graph, where |E | is even. In this paper we show that the line graph L(G) of G contains at least 2|E |−|V |+1 perfect matchings, and characterize G such that L(G) has exactly 2|E |−|V |+1 perfect matchings. As applications, we use a unified approach to solve the dimer problem on the Kagomé lattice, 3.12.12 lattice, and Sierpinski gasket with dimension 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 connected graph 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 any A = {a1 , a2 , . . . , as } ⊂ V (G) (resp., E1 = {e1 , e2 , . . . , et } ⊂ E (G)), let G − A or G − a1 − a2 − · · · − as (resp., G − E1 or G − 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 that 0 ≤ 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 in E ′ . 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 set of 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 e and f in L(G) are joined by exactly µG (e, f ) edges. It is clear that M (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 line graph is claw-free, the line graph of a connected graph with an even size has at least one perfect matching. Obviously, there exists a one–one correspondence between perfect matchings of L(G) and decompositions of the edge set of G into paths with two 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 quantum chemistry [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 in graphs) 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: [email protected] (F. Dong), [email protected], [email protected] (W. Yan), [email protected] (F. Zhang).
0166-218X/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.dam.2012.10.032
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(see for example [21–23]). Note that the plane Kagomé, 3.3.12 lattices and the Sierpinski gasket are the line graphs of some graphs, 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 for M (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 . This result immediately implies that M (L(G)) ≥ 2m−n+1 . This section also provides another lower bound for M (L(G)): if G is 2-connected, then M (L(G)) ≥ 52 (∆!)1/4 2m−n+1 , where ∆ is the maximum degree of G. 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 can be used to determine the number of perfect matchings of a wide class of graphs which are themselves line graphs of subcubic graphs (i.e., graphs of maximum degree at most 3). In particular, in Section 5, we use it to determine the number of perfect matchings of the Kagomé lattice, 3.12.12 lattice, and Sierpinski gasket with dimension two in the context of statistical physics. 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 thus M (L(T )) = 0. Hence, for determining M (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 a forest, 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). For any 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 )) =
n
p(T − vi )!!.
i=1
Two results follow immediately from Lemma 1. Corollary 1. Let T be a tree of order n, where n > 1 is odd. Then M (L(T )) ≥ 1, where the equality holds if and only if p(T −v) = 0 or 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 for M (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), is defined to be m − 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 expression for M (L(T )) for any tree T of odd order. In this section, we shall consider low bounds for M (L(G)) for all connected graphs G. We show that M (L(G)) ≥ 2c (G) for any connected graph G and M (L(G)) ≥ 25 (∆!)1/4 2c (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 any x ∈ V (G), let Ex be the set of edges in G which 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 perfect matching of a line graph corresponds to a decomposition of its edge set into paths with 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 ′ of m − 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 for i = 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 the graph obtained from G − E ′ by adding m − n + 1 new vertices w1 , w2 , . . . , wm−n+1 and for every i with 1 ≤ i ≤ m − n + 1, adding a new edge joining wi to vi if ji = 0 or to ui if ji = 1. Note that G(j1 , j2 , . . . , jm−n+1 ) is a tree of order m + 1 and size m. 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)) =
M (L(G(j1 , j2 , . . . , jm−n+1 ))),
(j1 ,j2 ,...,jm−n+1 )
where the summation ranges over all 2m−n+1 vectors (j1 , j2 , . . . , jm−n+1 ), where jk ∈ {0, 1}.
(1)
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By Corollary 3, the number of perfect matchings of the line graph of a connected graph G of order n and size m can be expressed as the summation of the numbers of perfect matchings of line graphs of 2m−n+1 trees of order m + 1. Note that, by Lemma 1, we can easily enumerate perfect matchings of the line graph of a tree. This results in an algorithm to enumerate perfect matchings of the line graph of a connected graph. We are now going to apply Corollaries 2 and 3 to get a lower bound for M (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 the equality 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 ))), where G(j1 , j2 , . . . , jm−n+1 ) is a tree of order m + 1 and size m. Thus M (L(G)) ≥ 2m−n+1 by Corollary 1. If ∆(G) ≤ 3, then each tree G(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, implying that M (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 ) defined in 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 isolated vertices, then M (L(G)) > 2m−n+1 by Corollaries 1 and 3. Remark 2. Little [8] proved that a graph G with vertex set V (G) has an even number of perfect matchings if and only if there is 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 graph G, 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 to an 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 line graphs 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 is not 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 thus M (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 a 2-connected graph G. For any integer r ≥ 0, define r η(r ) = (2k)!!. 2k 0≤k≤r /2 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 is connected. 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 graph obtained from G − x by adding s new vertices w1 , w2 , . . . , ws and s new edges joining ui to wi for all i = 1, 2, . . . , s. Applying Lemma 2 repeatedly yields that M (L(G)) =
M (L(GS ))M (L(T1,d(x)−|S | )),
S ⊆E x
where T1,t denotes the star with t + 1 vertices. Note that M (L(T1,t )) = t !! if t is even and M (L(T1,t )) = 0 otherwise. Thus M (L(G)) =
(2k)!!
0≤k≤d(x)/2
M (L(GS )).
S ⊆Ex |S |=d(x)−2k
As GS is a connected graph with n − 1 +|S | vertices and m − d(x)+|S | edges, we have M (L(GS )) ≥ 2m−n−d(x)+2 by Theorem 1. Thus M (L(G)) ≥
(2k)!!
0≤k≤d(x)/2
= 2m−n−d(x)+2
2m−n−d(x)+2
S ⊆Ex |S |=d(x)−2k
(2k)!!
0≤k≤d(x)/2
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 ) ≥
2r 5
(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 difficult to verify that η(r ) ≥ √2 (r + 2)! holds for r ≤ 10. For r > 10, this inequality can be proved inductively by applying the 30
recursive formula η(r ) = η(r − 1) + (r − 1)η(r − 2). The second inequality can be proved similarly.
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By Lemmas 3 and 4, we obtain the following lower bound for M (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 2 M (L(G)) ≥ √ ((∆ + 2)!)1/2 2m−n−∆+2 30
and
M (L(G)) ≥
2 5
(∆!)1/4 2m−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 second bound is better than the bound 2m−n+1 if and only if ∆ ≥ 5. However, both bounds in Theorem 2 are much better than 2m−n+1 when ∆ is large. 4. The extremal graphs G with M (L (G )) = 2m−n+1 In this section, we characterize those connected graphs G such that the equality M (L(G)) = 2m−n+1 holds. We first obtain the 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, then M (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 a subset 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 incident with 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 at least 3 vertices of degree 1, implying that M (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 equality holds 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 that M (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 consider the case that G is not 2-connected and find a necessary and sufficient condition for the equality M (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 of u 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 edge of 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) =
k (fGi (u) + gGi (u)), i=1
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) =
k k (fGi (u) + gGi (u)) ≡ |E (Gi )| = m(mod 2). i=1
i =1
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 of G, 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 holds for x ∈ {u, v}.
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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 not a 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 and gG (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)| and m = |E (G)|. Theorem 3. Let G be a connected graph with n vertices and m edges, where m is even. Then the following statements are equivalent: 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 that G − {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 of components 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 ) = ∅ or V (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 of G(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. So p(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 3 implies that M (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 have M (L(G)) = 1 = 2m−n+1 , contradicting the assumption on G. Thus G contains cycles. Let e be an edge in some cycle of G and u and v be its ends. Both G(u, w) and G(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 cyclomatic number 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 that M ((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 therefore fG (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 are themselves line graphs of sub-cubic graphs. In this section we apply this result to enumerate perfect matchings of a Sierpinski gasket 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 difficult to see that SG2 (n) has 32 (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.
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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 perfect n matchings of SG2 (n) equals 2(3 −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 ) is isomorphic 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 is easily verified that L(Gn ) ∼ = SG2 (n). By the definition of Gn , Gn has order 3n + 3 and size 23 (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, 3 n n n M (SG2 (n)) = M (L(Gn )) = 2 2 (3 +1)−(3 +3)+1 = 2(3 −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 edges
in 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 a dimer formulation of the Ising model. By means of Pfaffians, Fisher [5] and Wu [22] proved that the logarithm of the number of perfect matchings of RT (n, m), divided by 3(m + 1)(n + 1) (the number of edges of each of the perfect matchings of RT (n, m)), converges to 13 ln 2 as m, n → ∞, which is called the entropy of RT (n, m) by statistical physicists. We shall derive by 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 by statistical physicists [3,5,22]. The hexagonal lattice H T (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 , c1∗ ), (c1 , c2∗ ), . . . , (cn−1 , cn∗ ), (cn , d∗m+1 ) are edges in H T (n, m). It is not difficult to see that the line graph of S (H T (n, m)) is RT (n, m), where S (H T (n, m)) is the graph obtained from H T (n, m) by subdividing each edge of H T (n, m) once. Note that S (H T (n, m)) has order 5(m + 1)(n + 1), size 6(m + 1)(n + 1) and maximum degree 3. By Theorem 1, we have M (RT (n, m)) = M (L(S (H T (n, m)))) = 26(m+1)(n+1)−5(m+1)(n+1)+1 = 2mn+m+n+2 .
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a
b
Fig. 3. (a) The 3.12.12 lattice RT (n, m) with toroidal boundary condition. (b) The hexagonal lattice H T (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 vj∗ for all j = 1, 2, . . . , n, the
resulting 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 and Wang [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 a new 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 H T (n − 1, 2m − 1) in Fig. 3(b), which is also illustrated by dashed lines in Fig. 4. Note that H T (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(H T (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 also thank 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 the edge 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 the NSFC Grant (11171134) and the Foundation for Young Professors of Jimei University. The third author was supported by the NSFC Grant (10831001). References [1] S.-C. Chang, L.-C. Chen, Dimer coverings on the Sierpinski gasket, J. Stat. Phys. 131 (2008) 631–650. [2] M. Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry, J. Combin. Theory Ser. A 77 (1997) 67–97. [3] V. Elser, Nuclear antiferromagnetism in a registered 3 He solid, Phys. Rev. Lett. 62 (1989) 2405–2408.
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I. Fischer, C.H.C. Little, Even circuits of prescribed clockwise parity, Electron. J. Combin. 10 (2003) R45. M.E. Fisher, On the dimer solution of planar Ising models, J. Math. Phys. 7 (1966) 1776–1781. G.G. Hall, A graphic model of a class of molecules, Int. J. Math. Educ. Sci. Technol. 4 (1973) 233–240. M. Jerrum, Two-dimensional monomer–dimer systems are computationally intractable, J. Stat. Phys. 48 (1987) 121–134. C.H.C. Little, The parity of the number of 1-factors of a graph, Discrete Math. 2 (1972) 179–181. Z. Lonc, Edge decomposition into copies of sK1,2 is polynomial, J. Combin. Theory Ser. B 69 (1997) 164–182. L. Lovász, M. Plummer, Matching Theory, in: Ann. Discrete Math., vol. 29, North-Holland, New York, 1986. L. Pauling, The Nature of Chemical Bond, Cornell. Univ. Press, Ithaca, New York, 1939. A.J. Phares, F.J. Wunderlich, Thermodynamics and molecular freedom of dimers on plane honeycomb and Kagomé lattices, Nuovo Cimento B 101 (1988) 653–686. J. Propp, Enumerations of Matchings: Problems and Progress, New Perspectives in Geometric Combinatorics, in: MSRI Publications, vol. 38, Cambridge University Press, Cambridge, UK, 1999, pp. 255–291. N. Robertson, P.D. Seymour, R. Thomas, Permanents, Pfaffian, orientations and even directed circuits, Ann. of Math. 150 (1999) 929–975. A. Schrijver, Counting 1-factors in regular bipartite graphs, J. Combin. Theory Ser. B 72 (1998) 122–135. D.P. Sumner, Graphs with 1-factors, Proc. Amer. Math. Soc. 42 (1974) 8–12. R. Swinborne-Sheldrake, W.C. Herndon, I. Gutman, Kekulé structures and resonance energies of benzennoid hydrocarbons, Tetrahedron Lett. (1975) 755–758. N. Teypaz, C. Rapine, Graph decomposition intor paths under length constraints, BQR INPG’Optimisation du transport de fret par l’utilisation de plateformes logistiques, 2008. L.G. Valiant, The complexity of computing the permanent, Theoret. Comput. Sci. 8 (1979) 410–421. M.L. Vergnas, A note on matchings in graphs, Cah. Cent. Etud. Rech. Oper. 17 (1975) 257–260. Colloque sur la Theorie des Graphes (Paris, 1974). F. Wang, F.Y. Wu, Exact solution of close-packed dimers on the kagomé lattice, Phys. Rev. E 75 (2007) 040105(R). F.Y. Wu, Dimers on two-dimensional lattices, Int. J. Mod. Phys. B 20 (2006) 5357–5371. F.Y. Wu, F. Wang, Dimers on the kagomé lattice I: finite lattices, Physica A 387 (2008) 4148–4156.
Contents lists available at SciVerse ScienceDirect
Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
On the number of perfect matchings of line graphs Fengming Dong a , Weigen Yan b,∗ , Fuji Zhang c a
National Institute of Education, Nanyang Technological University, Singapore
b
School of Sciences, Jimei University, Xiamen 361021, China
c
School of Mathematical Science, Xiamen University, Xiamen 361005, China
article
info
Article history: Received 17 February 2011 Received in revised form 28 June 2012 Accepted 18 October 2012 Available online 11 November 2012 Keywords: Perfect matching Line graph Cyclomatic number
abstract Let G = (V , E ) be a connected graph, where |E | is even. In this paper we show that the line graph L(G) of G contains at least 2|E |−|V |+1 perfect matchings, and characterize G such that L(G) has exactly 2|E |−|V |+1 perfect matchings. As applications, we use a unified approach to solve the dimer problem on the Kagomé lattice, 3.12.12 lattice, and Sierpinski gasket with dimension 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 connected graph 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 any A = {a1 , a2 , . . . , as } ⊂ V (G) (resp., E1 = {e1 , e2 , . . . , et } ⊂ E (G)), let G − A or G − a1 − a2 − · · · − as (resp., G − E1 or G − 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 that 0 ≤ 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 in E ′ . 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 set of 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 e and f in L(G) are joined by exactly µG (e, f ) edges. It is clear that M (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 line graph is claw-free, the line graph of a connected graph with an even size has at least one perfect matching. Obviously, there exists a one–one correspondence between perfect matchings of L(G) and decompositions of the edge set of G into paths with two 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 quantum chemistry [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 in graphs) 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: [email protected] (F. Dong), [email protected], [email protected] (W. Yan), [email protected] (F. Zhang).
0166-218X/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.dam.2012.10.032
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(see for example [21–23]). Note that the plane Kagomé, 3.3.12 lattices and the Sierpinski gasket are the line graphs of some graphs, 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 for M (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 . This result immediately implies that M (L(G)) ≥ 2m−n+1 . This section also provides another lower bound for M (L(G)): if G is 2-connected, then M (L(G)) ≥ 52 (∆!)1/4 2m−n+1 , where ∆ is the maximum degree of G. 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 can be used to determine the number of perfect matchings of a wide class of graphs which are themselves line graphs of subcubic graphs (i.e., graphs of maximum degree at most 3). In particular, in Section 5, we use it to determine the number of perfect matchings of the Kagomé lattice, 3.12.12 lattice, and Sierpinski gasket with dimension two in the context of statistical physics. 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 thus M (L(T )) = 0. Hence, for determining M (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 a forest, 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). For any 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 )) =
n
p(T − vi )!!.
i=1
Two results follow immediately from Lemma 1. Corollary 1. Let T be a tree of order n, where n > 1 is odd. Then M (L(T )) ≥ 1, where the equality holds if and only if p(T −v) = 0 or 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 for M (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), is defined to be m − 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 expression for M (L(T )) for any tree T of odd order. In this section, we shall consider low bounds for M (L(G)) for all connected graphs G. We show that M (L(G)) ≥ 2c (G) for any connected graph G and M (L(G)) ≥ 25 (∆!)1/4 2c (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 any x ∈ V (G), let Ex be the set of edges in G which 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 perfect matching of a line graph corresponds to a decomposition of its edge set into paths with 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 ′ of m − 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 for i = 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 the graph obtained from G − E ′ by adding m − n + 1 new vertices w1 , w2 , . . . , wm−n+1 and for every i with 1 ≤ i ≤ m − n + 1, adding a new edge joining wi to vi if ji = 0 or to ui if ji = 1. Note that G(j1 , j2 , . . . , jm−n+1 ) is a tree of order m + 1 and size m. 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)) =
M (L(G(j1 , j2 , . . . , jm−n+1 ))),
(j1 ,j2 ,...,jm−n+1 )
where the summation ranges over all 2m−n+1 vectors (j1 , j2 , . . . , jm−n+1 ), where jk ∈ {0, 1}.
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By Corollary 3, the number of perfect matchings of the line graph of a connected graph G of order n and size m can be expressed as the summation of the numbers of perfect matchings of line graphs of 2m−n+1 trees of order m + 1. Note that, by Lemma 1, we can easily enumerate perfect matchings of the line graph of a tree. This results in an algorithm to enumerate perfect matchings of the line graph of a connected graph. We are now going to apply Corollaries 2 and 3 to get a lower bound for M (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 the equality 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 ))), where G(j1 , j2 , . . . , jm−n+1 ) is a tree of order m + 1 and size m. Thus M (L(G)) ≥ 2m−n+1 by Corollary 1. If ∆(G) ≤ 3, then each tree G(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, implying that M (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 ) defined in 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 isolated vertices, then M (L(G)) > 2m−n+1 by Corollaries 1 and 3. Remark 2. Little [8] proved that a graph G with vertex set V (G) has an even number of perfect matchings if and only if there is 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 graph G, 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 to an 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 line graphs 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 is not 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 thus M (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 a 2-connected graph G. For any integer r ≥ 0, define r η(r ) = (2k)!!. 2k 0≤k≤r /2 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 is connected. 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 graph obtained from G − x by adding s new vertices w1 , w2 , . . . , ws and s new edges joining ui to wi for all i = 1, 2, . . . , s. Applying Lemma 2 repeatedly yields that M (L(G)) =
M (L(GS ))M (L(T1,d(x)−|S | )),
S ⊆E x
where T1,t denotes the star with t + 1 vertices. Note that M (L(T1,t )) = t !! if t is even and M (L(T1,t )) = 0 otherwise. Thus M (L(G)) =
(2k)!!
0≤k≤d(x)/2
M (L(GS )).
S ⊆Ex |S |=d(x)−2k
As GS is a connected graph with n − 1 +|S | vertices and m − d(x)+|S | edges, we have M (L(GS )) ≥ 2m−n−d(x)+2 by Theorem 1. Thus M (L(G)) ≥
(2k)!!
0≤k≤d(x)/2
= 2m−n−d(x)+2
2m−n−d(x)+2
S ⊆Ex |S |=d(x)−2k
(2k)!!
0≤k≤d(x)/2
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 ) ≥
2r 5
(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 difficult to verify that η(r ) ≥ √2 (r + 2)! holds for r ≤ 10. For r > 10, this inequality can be proved inductively by applying the 30
recursive formula η(r ) = η(r − 1) + (r − 1)η(r − 2). The second inequality can be proved similarly.
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By Lemmas 3 and 4, we obtain the following lower bound for M (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 2 M (L(G)) ≥ √ ((∆ + 2)!)1/2 2m−n−∆+2 30
and
M (L(G)) ≥
2 5
(∆!)1/4 2m−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 second bound is better than the bound 2m−n+1 if and only if ∆ ≥ 5. However, both bounds in Theorem 2 are much better than 2m−n+1 when ∆ is large. 4. The extremal graphs G with M (L (G )) = 2m−n+1 In this section, we characterize those connected graphs G such that the equality M (L(G)) = 2m−n+1 holds. We first obtain the 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, then M (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 a subset 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 incident with 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 at least 3 vertices of degree 1, implying that M (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 equality holds 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 that M (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 consider the case that G is not 2-connected and find a necessary and sufficient condition for the equality M (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 of u 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 edge of 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) =
k (fGi (u) + gGi (u)), i=1
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) =
k k (fGi (u) + gGi (u)) ≡ |E (Gi )| = m(mod 2). i=1
i =1
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 of G, 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 holds for x ∈ {u, v}.
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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 not a 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 and gG (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)| and m = |E (G)|. Theorem 3. Let G be a connected graph with n vertices and m edges, where m is even. Then the following statements are equivalent: 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 that G − {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 of components 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 ) = ∅ or V (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 of G(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. So p(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 3 implies that M (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 have M (L(G)) = 1 = 2m−n+1 , contradicting the assumption on G. Thus G contains cycles. Let e be an edge in some cycle of G and u and v be its ends. Both G(u, w) and G(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 cyclomatic number 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 that M ((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 therefore fG (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 are themselves line graphs of sub-cubic graphs. In this section we apply this result to enumerate perfect matchings of a Sierpinski gasket 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 difficult to see that SG2 (n) has 32 (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.
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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 perfect n matchings of SG2 (n) equals 2(3 −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 ) is isomorphic 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 is easily verified that L(Gn ) ∼ = SG2 (n). By the definition of Gn , Gn has order 3n + 3 and size 23 (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, 3 n n n M (SG2 (n)) = M (L(Gn )) = 2 2 (3 +1)−(3 +3)+1 = 2(3 −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 edges
in 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 a dimer formulation of the Ising model. By means of Pfaffians, Fisher [5] and Wu [22] proved that the logarithm of the number of perfect matchings of RT (n, m), divided by 3(m + 1)(n + 1) (the number of edges of each of the perfect matchings of RT (n, m)), converges to 13 ln 2 as m, n → ∞, which is called the entropy of RT (n, m) by statistical physicists. We shall derive by 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 by statistical physicists [3,5,22]. The hexagonal lattice H T (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 , c1∗ ), (c1 , c2∗ ), . . . , (cn−1 , cn∗ ), (cn , d∗m+1 ) are edges in H T (n, m). It is not difficult to see that the line graph of S (H T (n, m)) is RT (n, m), where S (H T (n, m)) is the graph obtained from H T (n, m) by subdividing each edge of H T (n, m) once. Note that S (H T (n, m)) has order 5(m + 1)(n + 1), size 6(m + 1)(n + 1) and maximum degree 3. By Theorem 1, we have M (RT (n, m)) = M (L(S (H T (n, m)))) = 26(m+1)(n+1)−5(m+1)(n+1)+1 = 2mn+m+n+2 .
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a
b
Fig. 3. (a) The 3.12.12 lattice RT (n, m) with toroidal boundary condition. (b) The hexagonal lattice H T (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 vj∗ for all j = 1, 2, . . . , n, the
resulting 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 and Wang [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 a new 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 H T (n − 1, 2m − 1) in Fig. 3(b), which is also illustrated by dashed lines in Fig. 4. Note that H T (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(H T (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 also thank 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 the edge 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 the NSFC Grant (11171134) and the Foundation for Young Professors of Jimei University. The third author was supported by the NSFC Grant (10831001). References [1] S.-C. Chang, L.-C. Chen, Dimer coverings on the Sierpinski gasket, J. Stat. Phys. 131 (2008) 631–650. [2] M. Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry, J. Combin. Theory Ser. A 77 (1997) 67–97. [3] V. Elser, Nuclear antiferromagnetism in a registered 3 He solid, Phys. Rev. Lett. 62 (1989) 2405–2408.
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