Graphs and Combinatorics(1997) 13:295-304

Perfect Matchings of Polyomino Graphs*

Graphs andCombinatorics© Springer-Verlag 1997

Heping Zhang1 and Fuji Zhang?

1 Department of Mathematics, Lanzhou University, Lanzhou, Gansu 730000, P.R. China2 Department of Mathematics, Xiamen University, Xiamen, Fujian 361005, P.R. China

Abstract. This paper gives necessary and sufficient conditions for a polyomino graph tohave a perfect matching and to be elementary, respectively. As an application, we candecompose a non-elementary polyominowithperfectmatchingsinto a numberof elementarysubpolyominoes so that the number of perfect matchings of the original non-elementarypolyomino is equal to the product of those of the elementarysubpolyominoes.

1. Introduction

A polyomino (graph) is a connected finite subgraph of the infinite plane grid suchthat each interior face is surrounded by a regular square of side length 1 (called acell) and each edge belongs to at least one cell. A hexagonal system (or polyhexgraph) is a finite 2-connected plane graph such that each interior face is sur­rounded by a regular hexagon of side length 1. A perfect matching or I-factor of agraph G is a set of independent edges of G covering all vertices of G. An edge of agraph G is said to be allowed if it lies in some perfect matching of G and forbiddenotherwise. A connected graph G is said to be elementary if all its allowed edgesform a connected subgraph of G. It is known that a connected bipartite graph Giselementary if and only if each edge of it is allowed (i.e. G is l-extendable) [1]. Thisterminology is due to Lovasz and Plummer [1] and Hetyei [2].

In how many ways can a polyomino be fully covered by dominoes (1 x 2 rec­tangles)? In graph theoretic terms, this problem is to determine the number ofperfect matchings of its inner dual graph (the inner dual graph of a polyomino P isdefined as a plane graph in which the vertex-set is the set of all cells of P and twovertices are adjacent provided the corresponding two cells have an edge in com­mon) . A domino pattern of a polyomino P is obtained by paving or tiling domi­noes. There is a one-to-one correspondence between the domino patterns of P andperfect matchings of the inner dual of P, which is illustrated in Fig. 1. It turns outthat the perfect matching problem of polyominoes is closely related to dimerproblem in crystal physics [3, 4, 5]. Kasteleyn [3], a physicist, developed a so called

• Supported by NSFC

296 H. Zhang and F. Zhang

EE8A polyomino P a domino pattern of P a perfect matching

of the inner dual of P

Fig. 1. Correspondence between domino patterns of a polyomino and perfect matchings ofits inner dual

" Pfaffian method" and derived the explicit expression of the number of ways forcovering the m x n chessboard with mn even by dominoes. John et al. [5] andSachs [6] also considered the enumeration of perfect matchings of polyomi noes . Inaddition, Berge et al. [7] studied the general covering problem for polyominoes byintroducing hypergraphs. Various properties of polyominoes with perfect match­ings have been obtained, in particular one of the present authors [8] proved thatthe connectivity of Z-transformation graphs of perfect matchings of polyominoesis equal to the minimum degree with only two exceptions. It is natural to seeksimple necessary and sufficient conditions for the existence of perfect matchings ofpolyominoes. For hexagonal systems, Kostochka [10] and Zhang, Guo and Chen[9, 11] independently obtained certain necessary and sufficient conditions for theexistence of perfect matchings. For triangle graphs, Akiyama and Kano [12]obtained elegant results by using the concept of tough triangle graphs. In thispaper, the authors intend to solve the analogous problem for polyominoes.

2. General Bipartite Graphs

In this section, we first obtain some necessary and sufficient conditions for generalbipartite graphs to have a perfect matching that are somewhat stronger thanknown results.

Let G = (X , Y) denote a connected bipartite graph. We color properly verticesof G white and black. For any subgraph H of G, denote B(H) and W(H) the setsof black and white vertices of H , respectively. Hence (W(G), B( G)) is a bipartitionof G. Set A = IB(G)I-IW(G)I. For S £ V(G), denote by N(S) the neighbor setof S, and <S) the subgraph of G induced by S. Denote by IX I the cardinality of afinite set X.

We now list two basic results which can be found in the book of Lovasz andPlummer [1].

Theorem 2.1. Let G = (X , Y) be a bipartite graph. Then G has a perfect matchingif and only if A = 0 and IN(S)I ~ lSI for every S £ x.

Theorem 2.2. Let G = (X , Y) be a connected bipartite graph. Then G is elementaryif any only if A = 0 and IN(S) I > lSI for every proper subset S of X.

Perfect Matchings of Polyomino Graphs 297

It will be seen that these two theorems can lead to somewhat stronger results .

Definition 2.3. Let G be a connected bipartite graph. A subset EC of E( G) is calledan elementary edge-cut if it satisfies the following two statements:

(1) G - EC has exactly two components G) and G2, and(2) Each edge of EC is incident with a black vertex of G) and a white vertex of G2.

Let G) and G2 be two vertex-disjoint subgraphs of G. Denote by (G) , G2) theset of those edges with one end vertex in G) and the other in G2.

Remark 2.4. Let G be a connected plane bipartite graph with an elementary edge­cut EC of size greater than one. If EC contains an edge of a face-boundary, then itmust contain precisely two edges of it.

Theorem 2.5. Let G be a connected bipartite graph. Then G has a perfect matchingifand only if L1 = 0 and IB(Gdl ~ IW(G))I for every elementary edge-cut of G.

Proof We first prove the necessity. Suppose that G has a perfect matching. Obvi­ously, L1 = O. Let EC be any elementary edge-cut of G. Set W( Gd = S, thenN(S) = B(G)) by the connectedness of G). From Theorem 2.1, lSI ~ IN(S)I, i.e.IB(G))I ~ IW(Gdl·

We now prove the sufficiency. Let S s;; W (G). IfS = W (G), by our assumption,N(S) = B(G) and IN(S)I = lSI. So assume that S -=F W(G) . Let G) = (SUN(S)and G2= (V(G)\(SUN(S))) (the vertex-set of G2 is not empty) . It is easy to seethat each edge of the edge-cut (G), G2) is incident with a black vertex in G) and awhite vertex in G2.

Assume that G) is connected. Let G~, i = 1,2, . . . ,m, be the components of G2•

Put G{ = G - V( G4). Hence both G{ and G~ are connected, and (G{, G~) is anelementary edge-cut of G. By assumption, then IB(GDI ~ IW(GDI. Since L1 = 0,IB(G~)I ~ IW(G4) I and IW(G2)1= 2::;:)1 W(G4) I~ 2::;:) IB(G~)I =IB(G2)1. HenceIB(Gdl~IW(Gdl·

If G) is disconnected, we deal with each component of G) by the aboveapproach and the similar inequalities are valid . Taking the summation for bothsides of these inequalities we also have that IB(GI)I ~ IW(G))I. By Theorem 2.1,G has a perfect matching. 0

In the above proof, substituting strict inequalities for all correspondinginequalities, we immediately obtain

Theorem 2.6. Let G be a connected bipartite graph. Then G is elementary ifand onlyif L1 = 0 and IB(Gd I > I W (GI) 1 for every elementary edge-cut of G.

Let us now assume that G is a plane bipartite graph. By the boundary of G wealways mean the boundary of the exterior face of G. A boundary edge of G is onewhich lies on the boundary of G. The interior vertices of G are those which are notvertices of the exterior face of G.

298 H. Zhang and F . Zhang

Theorem 2.7. Let G be a connected plane bipartite graph each interior vertexof which has the same degree. Then

(1) G has a perfect matching ifand only if L1 = 0 and IB(G))I ~ IW(G))lfor everyelementary edge-cut of G containing a boundary edge of G,

(2) G is elementary if and only if L1 = 0 and IB(G))I > IW(G))I for every elemen-tary edge-cut of G containing a boundary edge of G.

Proof We adopt the notation in the proof of Theorem 2.5. If an elementary edge­cut does not contain boundary edges of G, then all the vertices of G) or G2 (say G))must be interior vertices of G. It is obvious that IE(Gdl = dl W(Gdl = dIB(G)) I­1(G), G2)I, where d is the degree of the interior vertices of G. Since (G), G2) =I- 0,we have that IB(Gdl > 1W(Gdl. Moreover by Theorems 2.5 and 2.6 the theoremfollows. 0

Conclusion (1) of Theorem 2.7 is actually a generalization of a related result inhexagonal systems [9].

As an immediate consequence, we have

Corollary 2.8. Let G be a connected plane bipartite graph each interior vertex ofwhich has the same degree. Assume that G has a perfect matching. Then G has aforbidden edge if and only if there exists an elementary edge-cut EC containing aboundary edge of G such that each edge ofEC is forbidden.

Proof It suffices to prove the necessity. Suppose that G has a forbidden edge. ByTheorem 2.7(2), we have that there is an elementary edge-cut EC of G containinga boundary edge of G such that IB(Gdl ~ 1W(Gdl . On the other hand, by Theo­rem 2.5, it follows that IB(Gdl ~ 1W(Gd. Thus IB(Gdl = I W(Gdl, which impliesthat each edge of EC = (G) , G2) is forbidden. 0

In order to get a simple criterion for determining which plane bipartite graphsare elementary, further terminology is needed. Let G be a plane bipartite graphwith a perfect matching M . A cycle C of G is called an M-alternating cycle if allthe edges of C appear alternately in M and E(G)\M. A cycle C of G is said to benice if G has a perfect matching M such that C is an M-alternating cycle. Thefollowing useful result was recently obtained by the authors [13].

Lemma 2.9. Let G be a plane graph with more than one edge. Then G is an ele­mentary bipartite graph ifand only ifeach face -boundary of G is a nice cycle.

We now immediately obtain the following main result.

Theorem 2.10. Let G be a connected plane bipartite graph each interior vertexofwhich has the same degree. Then G is elementary ifand only if the boundary of Gis a nice cycle.

Proof If G is elementary, by Lemma 2.9 it follows that the boundary of G is a nicecycle. Conversely, suppose that the boundary of G is a nice cycle, which implies

Perfect Matchings of Polyomino Graphs 299

that each edge of the boundary of G is allowed. If G is not elementary , by Corol­lary 2.8, we immediately know that G must have a forbidden edge lying on theboundary of G, a contradiction. 0

It is worth pointing out that Theorem 2.10 can provide a fast algorithm forrecognizing whether graphs of this kind (connected plane bipartite graphs in whichall interior vertices have the same degree) are elementary, which is outlined asfollows. Because elementary bipartite graphs are either 2-connected graphs or K2(the complete graph of two vertices), we only consider 2-connected plane bipartitegraphs G the interior vertices of which are of the same degree. First delete all thevertices of the boundary of G together with their incident edges. Then checkwhether or not the remainder has a perfect matching.

Furthermore, Lemma 2.9 and Theorem 2.10 can be used to produce directlythe following known basic results in polyhex and polyomino graphs.

Corollary 2.ll [14]. Let H be a polyhex (i.e. a hexagonal system). Then the follow­ing statements are equivalent.

(i) H is normal (i.e. elementary),(ii) the boundary ofH is a nice cycle ,(iii) each unit hexagon ofH is a nice cycle.

Proof (i){:} (ii). If each unit hexagon of H is a nice cycle, it is easy to see that eachedge of H, which belongs to at least one unit hexagon, is allowed and hence H isnormal. If H is normal, by Lemma 2.9 it follows that each hexagon of H is a nicecycle.

(i){:} (ii). Since all interior vertices of a hexagonal system have the same degree 3,Theorem 2.10 implies that H is normal if and only if the boundary of H is a nice~~. 0

By the same arguments, we have that

Corollary 2.12 [8]. Let G be a polyomino. Then the following statements areequivalent.

(i) G is elementary,(ii) the boundary of G is a nice cycle ,(iii) each unit square of G is a nice cycle.

3. Polyomino Graphs

Let us introduce two definitions as follows.

Definition 3.1. Let G be a polyomino graph. A straight line segment C with endpoints PI and P2 is called a cut segment of G if

300

Fig. 2. A cut segment

H. Zhang and F. Zhang

Fig. 3. A g-eut segment

(l ) each of Pi and P2 is the center of an edge lying on the boundary of G,(2) P, P2 and all edges of G form an angle of~, and(3) the subgraph obtained from G by deleting all edges intersected by C has

exactly two components (see Fig. 2)

Definition 3.2. Let G be a polyomino. A broken line segment C = P\P2P3 is calleda g-cut segment of G if

(1) angle LP\P2P3 =~'

(2) each of P, and P3 is the center of an edge lying on the boundary of G,(3) P2 is the center of some edge e which is the bisector of the right angle

LP\P2P3, and(4) the subgraph obtained from G by deleting all edges intersected by C has

exactly two components (see Fig . 3).

The sets of edges of G intersected by cut and g-cut segments of G are calledcuts and g-cuts of G, respectively , and are denoted by CC. Obviously, these cuts andg-cuts of G are two special types of elementary edge-cuts of G. Below we will seethat for polyomino graphs these two special types of elementary edge-cuts playasignificant role.

We now state our main results as follows .

Theorem 3.3. Let G be a polyomino. Then G has a perfect matching if and only ifL1 = 0 and IB(Gt}I~ IW( Gt}I for every cut and g-cut of G.

Theorem 3.4. Let G be a polyomino. Then G is elementary ifand only if L1 = 0 andIB(Gt}I> IW( Gt}Ifor every cut and g-cut of G.

Before proving the theorems, let us make a remark. To ask only that the cutsegments alone, or the g-cut segments alone, satisfy that IB(G\) I~ IW( G\) I is notenough for our conclusion. For example, for the polyomino shown in Fig. 4,although L1 = 0 and all the g-cut segments fulfil the condition of Theorem 3.3, thecut segment P\P2 violates it; whereas, for a polyomino shown in Fig. 5, it is easilyverified that L1 = 0 and all the cut segments fulfil the condition of Theorem 3.3, butthe g-cut segment P\P2P3 violates it. From Theorem 2.7(1), these two poly­ominoes have no perfect matchings.

Perfect Matchings of Polyomino Graphs

Fig. 4

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A. .1 ... ,J...

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Fig. 5

301

Proof of Theorem 3.3. By Theorem 2.5, We only need to prove the sufficiency.Assume that L1 = 0 and IB(Gdl ~ IW(Gdl for every cut and g-cut segment of G.Since the interior vertices of G are of the same degree 4, by Theorem 2.7(1), itsuffices to prove that IB(GI)I ~ IW(Gdl (equivalently IB(G2)1 ~ IW(G2)1) forevery elementary edge-cut EC of G containing a boundary edge of G.

Let GI and G2 be two components of G - EC as described as in Definition 2.3.Since each edge belongs to at least one cell, the graph G is 2-edge connected andhence IEq > 1. By Remark 2.4 we can connect all the edges of EC with a brokenline segment (BLS) L = hiha . . . hn+lhn+2(n ~ 0) satisfying that

(1) each of hi and hn+2 is the center of some boundary edge of G,(2) hihi+ I , i = 1, 2, . .. , n + 1, and one edge direction of G form an angle oq,(3) hi, i = 2,3, . . . , n + 1, is the center of some edge of G not lying on the boundary

of G such that the edge is the bisector of the right angle Lhi-Ihihi+l ,(4) every point of the BLS L is either an interior or boundary point of some cell

(unit square) of G, and(5) the set of edges of G intersected by L is just the elementary edge-cut EC of G.

Obviously the number of turning points of the BLS L is n. For each turningpoint hi(2 ~ i ~ n + 1), the directed angle Lhi-lhihi+1 is oq or -~' We now shallshow that IB(GI)I ~ IW(Gdl by induction on the number n of turning points of L.For n = 0 or 1, since the BLS is just a so called "cut or g-cut segment" of G andthe EC is just a cut or g-cut of G, it follows that IB(Gdl ~ IW(Gdl by theassumption at the beginning of the proof.

In what follows, we assume that n ~ 2. There are two cases Lhlh2h3 =Lh2h3h4 and Lh1h2h3 = - Lh2h3h4 to be considered .

Let us now consider the first case. Without loss of generality, assume thatLhlh2h3 = Lh2h3h4 = - ~ and GI lies on the right side of the BLS L =hlh2 . . . hn+2. In order to simplify our discussion, a polyomino G in question isdrawn in the plane such that the turning points h2 and h3 of L are the centers ofhorizontal and vertical edges respectively (see Fig. 6). The points of a segment hih2intersected by a series of horizontal edges of the infinite plane grid are representedby h~ (= h2),hl, ...; while the points of a segment h3h4 intersected by a series of

302 H. Zhang and F. Zhang

Fig. 6. A broken line segment (BLS)L = h,h z . . .hg, n = 6

vertical edges are represented by hg(= h3),hl, . .. (see Fig. 6). The h~hrs aremutually parallel segments, which can be obtained by shifting hzh3• In order ofincreasing superscripts, let hih'{/ be the first such segment parallel to hahs andintersected by the boundary of G or a segment hjhj+1(j;;:: 4) of L. Let GJ be thesubgraph of GI induced by the vertices lying in the interior region of the closedbroken line segment hihzh3h'{/hi . Let Gl, Gf, .. . , Gf, HI, HZ, . . . ,H S (s or t;;:: 1)be the components of the subgraph obtained from GI by deleting all the vertices ofG? together with their incident edges such that Gi(1 ::;; i ::;; i + 1) contains aboundary edge of G and all the vertices of H i are interior vertices of G. It is easy tosee that IB(G?)I;;::IW(G?)I. By the proof of Theorem 2.7, we have that IB(Hi)l;;::IW(Hi)l,i= l, ,s. Put G~ = G- V(GD,i= 1, ... .t. It is easily seen that the(Gi , G~) , i = 1, , t, are the elementary edge-cuts which correspond to BLS' ofs;n - 1 turning points . By induction hypothesis, it follows that IB(GDI ;;:: IW( GDI.Hence

IB(Gd l-IW(Gdlt S

= ~)IB(GDI-IW(G:)I) + ~)IB(Hi)I-IW(Hi)l)i=O i= 1

;;::0

So IB(Gdl;;::IW(GI)I.In the case that Lhlhzh3 = - Lhahsh« (see Fig. 7), by the similar arguments

we can deduce the same conclusion. The proof is complete. 0

Perfect Matchings of Polyomino Graphs

Fig. 7

The proof of Theorem 3.4 is similar to the above.Combining Theorems 3.3 and 3.4 we have

303

Corollary 3.5. Let G be a polyomino with a perfect matching. Then G has a for­bidden edge ifand only if there exists a cut or g-cut re of G such that each edge ofreis a forbidden edge.

Assume that G is a non-elementary polyomino with perfect matchings . ByCorollary 3.5 it is easy to see that the subgraph obtained from G by deleting allforbidden edges of G consists of isolated edges and at least two components withmore than two vertices which are elementary subpolyominoes of G (called ele­mentary components of G). Then we have the following decomposition theorem.

Theorem 3.6. Let G be a non-elementary polyomino with perfect matchings. Denoteby <1>( G) the number ofperfect matchings of G. Then

s

<1>( G) = II <1>( Gi ) ,

i=1

where Gi , i = 1,2, . .. . s, are the elementary components of G.We conclude this paper with the following open problem.

Open Problem 3.7. Is there a linear algorithm to produce a perfect matching ofpolyominoes?

Remark 3.8. A linear algorithm for a perfect matching for hexagonal systems hasbeen given by Hansen and Zheng [15, 16].

Acknowledgement. We are grateful to the referee for many helpful suggestions.

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crete Appl. Math. 30, 63-75 (1991)15. Zheng, Maolin : Perfect matchings in benzenoid systems, Ph.D.thesis, Rutgers Uni­

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Received: January 12, 1994Revised: December 11, 1996