Factor-Critical Graphs with Given Number of Maximum Matchings

Digital Object Identifier (DOI) 10.1007/s00373-008-0816-6Graphs and Combinatorics (2008) 24:563–569

Graphs andCombinatorics© Springer-Verlag 2008

Factor-Critical Graphs with Given Number of MaximumMatchings�

Yan Liu1, Guiying Yan2

1 School of Mathematics, South China Normal University, Guangzhou, 510631 P.R. China.E-mail: luky [email protected]

2 Academy of Mathematics and Systems Science, Chinese Academy of Science,Beijing, 100080, P.R. China.

Abstract. A connected graph G is said to be factor-critical if G − v has a perfect matchingfor every vertex v of G. In this paper, the factor-critical graphs G with |V (G)| maximummatchings and with |V (G)| + 1 ones are characterized, respectively. From this, some specialbicritical graphs are characterized.

Key words. Maximum matching, factor-critical graph, bicritical graph.

1. Introduction and Terminology

The reader is referred to [1] and [2] for undefined terms and concepts. We shallconsider finite, undirected and simple graphs only. Let G be a graph. We say thata path of G is pending if each of its interior vertices has degree 2 in G. A cycle isodd if the length of the cycle is odd. A subgraph H of G is a block of G if H is acut edge or H is a maximal 2-connected subgraph. Then for any block H of G, Gcan be represented as G = H + H2 + · · · + Hc, such that Hi is a block of G, and Hi

has exactly one common vertex with Gi−1 for 2 ≤ i ≤ c, where c is the number ofblocks of G, G1 = H and Gi = Gi−1 + Hi .

A theta graph �(l1, l2, l3) is a graph consisting of three pairwise internallydisjoint paths with common end-vertices and lengths l1, l2, l3, respectively. Then�(l1, l2, l3) has l1 + l2 + l3 − 1 vertices. We say that a connected graph G is f actor -cri tical if G − v has a perfect matching for every vertex v ∈ V (G). Then a factor-critical graph G has an odd number of vertices and its minimum degree is at leasttwo. Therefore, we have the following.

Proposition 1.1. Let G be a factor-critical graph with |V (G)| > 1. Then |E(G)| ≥|V (G)| and equality holds if and only if G is an odd cycle.

∗ This work is supported by the Ph.D. Programs Foundation of Ministry of Education ofChina (No.20070574006 ) and the NNSF(10201019) of China.

564 Y. Liu and G. Yan

Proposition 1.2. Let G be a theta graph �(l1, l2, l3). Then G is factor-critical if andonly if one of l1, l2, l3 is even and the others are odd.

Let G be a connected graph, A the set of cut vertices of G and B the set of blocks ofG. We construct a bipartite graph CB(G) with bipartition (A, B) such that v ∈ Ais adjacent to G ′ ∈ B in CB(G) if and only if v is a vertex of G ′. We call CB(G) thecut-block graph of G. The following can be verified.

Proposition 1.3. [5] Let G be a connected graph. Then CB(G) is a tree.

The problem of finding the number of maximum matchings of a graph plays animportant role in graph theory and combinatorial optimization since it has a widerange of applications. For example, in the chemical context, the number of perfectmatchings of bipartite graphs is referred to as Kekule structure count [3,4].

A graph G with at least one edge is said to be bicri tical if G −x − y has a perfectmatching for any two vertices x and y. Bicritical graphs are an important kind ofgraphs with a perfect matching. According to the definition of bicritical graphs, itis clear that G is a bicritical graph if and only if for every vertex v of G, G − v isfactor-critical. Then we have that the number of perfect matchings of a bicriticalgraph G is equal to the number of maximum matchings of G − u if u is a vertex ofdegree |V (G)| − 1.

In this paper, the factor-critical graphs G with |V (G)| maximum matchings andwith |V (G)|+1 ones are characterized, respectively. From this, some special bicriticalgraphs are characterized.

2. Results and Proofs

The following lemmas are useful.

Lemma 2.1. [6] Let G be a factor-critical graph. Then G has at least |E(G)| − c + 1maximum matchings, where c is the number of blocks of G.

Lemma 2.2. [6] Let G be a factor-critical graph with c blocks. Then G has exactly|E(G)| − c + 1 maximum matchings if and only if every block G ′ of G is an odd cycleor G ′ can be represented as G ′ = C + P1 + · · · + Pk , where C is an odd cycle, Pi isa path of G ′ with odd length and having both end-vertices–but no interior vertices–inGi−1 and two end-vertices of Pi are connected by a pending path of G with length 2for each i , 1 ≤ i ≤ k, where G0 = C , and Gi = C + P1 + · · · + Pi .

Lemma 2.3. [2] Let G be a graph. Then G is factor-critical if and only if every blockof G is factor-critical.

Lemma 2.4. Let G be a theta graph �(l1, l2, l3) satisfying that l1 is even and l2, l3 areodd. Then G has exactly |V (G)| + l1

2 maximum matchings.

Factor-Critical Graphs with Given Number of Maximum Matchings 565

Fig. 1. Theta graph �(4, 3, 3).

Proof. Suppose that P1 = uv1 · · · vl1−1v, P2 = uu1 · · · ul2−1v and P3 = uw1 · · · wl3−1v

are three paths of G joining the end-vertices u and v(see Fig. 1). Then |V (G)| =l1 + l2 + l3 − 1 and G is factor-critical by Proposition 2.4. Let

M = {M |M is a maximum matching of G},Mi = {M ∈ M|Mmisses vi } for 1 ≤ i ≤ l1 − 1,

M′j = {M ∈ M|M misses u j } for 1 ≤ j ≤ l2 − 1,

M∗k = {M ∈ M|M misses wk} for 1 ≤ k ≤ l3 − 1,

M′ = {M ∈ M|M misses u or v}

Then

|M| =∑

1≤i≤l1−1

|Mi | +∑

1≤ j≤l2−1

|M′j | +

∑

1≤k≤l3−1

|M∗k | + |M′|

Simple checks show that

|Mi | = 2 for each odd i, for 1 ≤ i ≤ l1 − 1,

|Mi | = 1 for each even i, for 1 ≤ i ≤ l1 − 1,

|M′j | = 1, |M∗

k | = 1, |M′| = 2.

Therefore,

|M| = l1 − 1 + l12

+ l2 − 1 + l3 − 1 + 2 = |V (G)| + l12

.

The proof is completed. �

Lemma 2.5. [7] Let G be a path with 2k − 1 vertices. Then G has exactly k maximummatchings.

In the following, for any graph G, the number of maximum matchings of G isdenoted by m(G).

Lemma 2.6. Let G1 be a factor-critical graph and G a graph obtained from G1 byadding an odd cycle C which has exactly one common vertex with G1, say v0. ThenG is factor-critical and has exactly m(G1) + (|V (C)| − 1)mv0 maximum matchings,where mv0 is the number of perfect matchings of G1 − v0.


Proof. By Lemma 2.3, G is factor-critical as C is a block of G and C is factor-critical.Let C = v0v1 · · · v2k and V (G1) ∩ V (C) = {v0}. Let

M = {M |M is a maximum matching of G},M1 = {M ∈ M|M misses v0},M2 = {M ∈ M|v0 is matched with a vertex of G1 by M},M3 = {M ∈ M|v0v1 ∈ M}.M4 = {M ∈ M|v0v2k ∈ M}.

Then |M| = |M1|+|M2|+|M3|+|M4| and |M1|+|M2| = m(G1). By Lemma 2.5,C − v0 − v1 and C − v0 − v2k have k maximum matchings, each. Since G1 is factor-critical, G1 − v0 has a perfect matching. Simple checks show that for any maximummatching M ∈ M3, M consists of a perfect matching of G1 − v0 and a maximummatching of C − v0 − v1. Then |M3| = kmv0 . By the same reason, |M4| = kmv0 .Therefore,

|M| = m(G1) + 2kmv0 = m(G1) + (|V (C)| − 1)mv0 .

The proof is completed. �

Theorem 2.7. Let G be a factor-critical graph with c blocks. Then m(G) = |V (G)| ifand only if every block of G is an odd cycle.

Proof. Suppose that every block of G is an odd cycle. Then |E(G)|−c+1 = |V (G)|.By Lemma 2.2, m(G) = |E(G)| − c + 1. Hence m(G) = |V (G)|.

Conversely, suppose that m(G) = |V (G)|. By Lemma 2.1, m(G) ≥ |E(G)|−c+1.Then

|V (G)| ≥ |E(G)| − c + 1.

Hence

|E(G)| ≤ |V (G)| + c − 1.

Let v1, v2, . . . , vk be all cut vertices of G. Further, let d j be the number of blockscontaining v j for each j , 1 ≤ j ≤ k and CB(G) the cut-block graph of G. ThendC B(G)(v j ) = d j . By Proposition 1.3, CB(G) is tree. Then

E(C B(G)) = c + k − 1 =k∑

j=1

dC B(G)(v j ) =k∑

j=1

d j .

It follows that

k∑

j=1

(d j − 1) = c − 1.


Fig. 2. the factor-critical graph with |V (G)| + 1 maximum matchings.

By Lemma 2.3 , every block G ′ of G is factor-critical. Then G ′ has an odd numberof vertices and dG ′(v) ≥ 2 for each v ∈ V (G ′). It follows that dG(v) ≥ 2 for eachnon-cut vertex v ∈ V (G) and dG(v j ) ≥ 2d j for each j , 1 ≤ j ≤ k. Therefore,

|E(G)| = 12

∑

v∈V (G)

dG(v) ≥ |V (G)| +k∑

j=1

(d j − 1) = |V (G)| + c − 1.

Thus |E(G)| = |V (G)| + c − 1. This implies that dG(v) = 2 for each non-cut vertexv ∈ V (G) and dG(v j ) = 2d j for each cut vertex v j . Thus every block of G is an oddcycle. �

Theorem 2.8. Let G be a factor-critical graph with c blocks. Then m(G) = |V (G)|+1if and only if all blocks of G are odd cycles but one, say H1 and H1 is a theta graph�(l1, l2, l3) satisfying that l1 = 2 and the path of length l1 in H1 is a pending path ofG (see Fig. 2).

Proof. Suppose that all blocks of G are odd cycles but one block H1 and H1 is atheta graph �(l1, l2, l3) satisfying that l1 = 2 and the path of length l1 in H1 is apending path of G. Then

|E(G)| − c + 1 = |V (G)| + 1.

Since G is factor-critical, H1 is factor-critical by Lemma 2.3. Then l2 and l3 are oddby Proposition 2.4. Hence H1 can be represented as H1 = C + P , where C is an oddcycle with length l1 + l2, P is a path with length l3 and having both ends –but nointerior vertices–in C , and the ends of P are connected in C by a pending path of Gwith length 2. By Lemma 2.2, m(G) = |E(G)| − c + 1. Thus m(G) = |V (G)| + 1.

Conversely, suppose that m(G) = |V (G)|+1. By Lemma 2.1, m(G) ≥ |E(G)|−c + 1. Then

|V (G)| + 1 ≥ |E(G)| − c + 1. (1)

Let H1, H2, . . . , Hc be the blocks of G. Then

|V (G)| = |V (H1)| +∑

2≤i≤c

(|V (Hi )| − 1) =∑

1≤i≤c

|V (Hi )| − c + 1. (2)


By Lemma 2.3, every Hi is factor-critical. Then |E(Hi )| ≥ |V (Hi )| and the equalityholds if and only if Hi is an odd cycle by Proposition 1.1. Since G has exactly|V (G)| + 1 maximum matchings, there exists a block of G, say H1, which is not acycle by Theorem 2.7. Then |E(H1)| ≥ |V (H1| + 1. Hence, due to (2),

|E(G)| − c + 1 =∑

1≤i≤c

|E(Hi )| − c + 1 ≥∑

1≤i≤c

|V (Hi )| − c + 2=|V (G)| + 1. (3)

Due to (1) and (3), |V (G)|+1 = |E(G)|−c+1. It follows that |E(H1)| = |V (H1)|+1and |E(Hj )| = |V (Hj )| for each j , 2 ≤ j ≤ c.

Then Hj is an odd cycle. Let Hj = C j . Therefore, we can assume that G =H1 + C2 + C3 + · · · + Cc, where G1 = H1, and Ci has exactly one common vertexwith Gi−1 = H1 + C2 + · · · + Ci−1 for 2 ≤ i ≤ c.

Now we show that H1 is a theta graph. Since |E(H1)| = |V (H1)|+1 and dH1(v) ≥2 for any v ∈ V (H1), either there exists exactly one vertex of degree 4 and othervertices have degree 2, or there exist exactly two vertices of H1 of degree 3 and othervertices have degree 2. Further, since H1 is a block, H1 has exactly two vertices ofdegree 3, say u and v, and other vertices have degree 2. (If there exists exactly onevertex of degree 4 and other vertices have degree 2, the vertex of degree 4 is a cutvertex of G, a contradiction.) It follows that G1 is a theta graph �(l1, l2, l3) withcommon end-vertices u and v.

Now we show that l1 = 2. Since H1 is factor-critical, one of l1, l2, l3 is even andthe others are odd. W.L.O.G., suppose that l2 and l3 are odd, l1 is even. Then l1 ≥ 2.By Lemma 2.6,

m(G) ≥ m(Gc−1) + |V (Cc)| − 1 ≥ · · · ≥ m(H1) +∑

2≤i≤c

(|V (Ci )| − 1).

By Lemma 2.4,

m(H1) = |V (H1)| + l12

.

Due to (2), m(G) ≥ |V (G)| + l12 . Since m(G) = |V (G)| + 1, l1 ≤ 2. Thus l1 = 2.

Therefore,

m(H1) = |V (H1)| + 1.

Suppose that the path with length l1 in H1 is P1 = uwv.Now we show that dG(w) = 2. Suppose, to the contrary, that dG(w) ≥ 3. Since

H1 is a block of G and dH1(w) = 2, w is a cut vertex of G. Therefore, we canassume that G = H1 + C2 + · · · + Cc such that dH1(w) = 2 and dG2(w) ≥ 3, whereG2 = H1 + C2. Then w is the common vertex of H1 and C2. Clearly, H1 − w is aneven cycle. Hence H1 − w has two perfect matchings. By Lemma 2.6,

m(G2) ≥ m(H1) + 2(|V (C2)| − 1) = |V (H1)| + 2|V (C2)| − 1.


Further, by Lemma 2.6, m(Gi+1) ≥ m(Gi ) + |V (Ci+1)| − 1. Due to (2) and sinceC2 is odd,

m(G) ≥ m(G2) + (|V (C3)| − 1) + · · · + (|V (Cc)| − 1)

≥ |V (H1)| +∑

2≤i≤c

(|V (Ci )| − 1) + |V (C2)|=|V (G)| + |V (C2)|≥|V (G)| + 3.

This contradicts m(G) = |V (G)| + 1. The proof is completed. �

We know that the number of perfect matchings of a bicritical graph G is equalto the number of maximum matchings of G − u, when u ∈ V (G) with dG(u) =|V (G)| − 1. So we give the following results without proofs.

Corollary 2.9. Let G be a a bicritical graph with the maximum degree |V (G)|−1 and|V (G)| ≥ 6. Then G has exactly |V (G)| − 1 perfect matchings if and only if thereexists a unique vertex u of degree |V (G)| − 1 and every block of G − u is an odd cycle.

Corollary 2.10. Let G be a a bicritical graph with the maximum degree |V (G)| − 1and |V (G)| ≥ 6. Then G has exactly |V (G)| perfect matchings if and only if thereexists a unique vertex u of degree |V (G)| − 1 and all blocks of G − u are odd cyclesbut one, say H1 and H1 is a theta graph �(l1, l2, l3) with common endvertices x and ysatisfying that l1 = 2 and the path of length l1 in H1 joining x and y is a pending pathof G − u.

References

1. Bondy, J.A., Murty, U.S.R.: Graph theory with applications. Macmillan Press Ltd., London(1976)

2. Lovasz, L., Plummer, M.D.: Matching Theory. Elsevier Science Publishers, B. V. NorthHolland (1985)

3. Cyvin, S.J., Gutman, I.: Kekule structures and their symmetry properties. ComputersMath. Applic., 12-B (3/4), 859–876 (1986)

4. Brown, R.L.: Counting of resonance structures for large benzenoid polynuclear hydro-carbons. J. Comp. Chem. 4, 556–562 (1983)

5. Bollobas, B.: Extremal Graph Theory. Academic Press INC., New York (1978)6. Liu, Y., Hao, J.: Enumeration of near-perfect matchings of factor-critical graphs. Disc.

Math. 243, 259–266 (2002).7. Liu, Y., Lin, Y., Wang, S.: On the number of near-perfect matchings of a type of bigraphs,

Study Math., 2, 73–86 (1998)

Received: January 8, 2008Final Version received: September 27, 2008

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Factor-Critical Graphs with Given Number of Maximum Matchings