aa r X i v : . [ m a t h . C O ] F e b On the restricted matching of graphs in surfaces ∗ Qiuli Li, Heping Zhang † School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, P. R. ChinaE-mail addresses: [email protected], [email protected]
Abstract
A connected graph G with at least 2 m + 2 n + 2 vertices is said to have property E ( m, n ) if, for any two disjoint matchings M and N of size m and n respectively, G has a perfect matching F such that M ⊆ F and N ∩ F = ∅ . In particular, a graphwith E ( m,
0) is m -extendable. Let µ (Σ) be the smallest integer k such that no graphsembedded on a surface Σ are k -extendable. Aldred and Plummer have proved thatno graphs embedded on the surfaces Σ such as the sphere, the projective plane, thetorus, and the Klein bottle are E ( µ (Σ) − , G embedded ona surface has sufficiently many vertices, then G has no property E ( k − ,
1) for eachinteger k ≥
4, which implies that G is not k -extendable. In the case of k = 4, we getimmediately a main result that Aldred et al. recently obtained. Keywords:
Perfect matching; Restricted matching; Extendability; Graphs in surface.
AMS 2000 subject classification: A matching of a graph G is a set of independent edges of G and a matching is called perfect if it covers all vertices of G . A connected graph G with at least 2 m + 2 n + 2 vertices is said tohave property E ( m, n ) (or abbreviated as G is E ( m, n )) if, for any two disjoint matchings M , N ⊆ E ( G ) of size m and n respectively, there is a perfect matching F such that M ⊆ F and N ∩ F = ∅ . It is obvious that a graph with E (0 ,
0) has a perfect matching. Since properties E ( m,
0) and m -extendability are equivalent, property E ( m, n ) is somewhat a generalization ∗ This work was supported by NSFC (grant no. 10831001). † The Corresponding author. m -extendability. The concept of m -extendable graphs was gradually evolved from thestudy of elementary bipartite graphs and matching-covered graphs (i.e. each edge belongsto a perfect matching ) and introduced by M.D. Plummer [13] in 1980. For extensive studieson m -extendable graphs, see two surveys [9] and [10]. A basic property is stated as follows. Lemma 1.1. ([13]) Every m -extendable graph is ( m + 1) -connected. For a vertex v of a graph G , let N ( v ) denote the neighborhood of v , i.e., the set of verticesadjacent to v in G , and G [ N ( v )] the subgraph of G induced by N ( v ). Lemma 1.2. ([4]) Let v be a vertex of degree m + t in an m -extendable graph G . Then G [ N ( v )] does not contain a matching of size t . Porteous and Aldred [15] introduced the concept of property E ( m, n ) and focussed onwhen the implication E ( m, n ) → E ( p, q ) does and does not hold. From then on, the possibleimplications among the properties E ( m, n ) for various values of m and n are studied in[6, 14, 15]. The following three non-trivial results will be used later. Lemma 1.3. ([15]) If a graph G is E ( m, n ) , then it is E ( m, . Lemma 1.4. ([15]) If a graph G is E ( m, n ) , then it is E ( m − , n ) . Lemma 1.5. ([15]) If a graph G is E ( m, for m ≥ , then it is E ( m − , . The converse of Lemma 1.5 does not hold. For example, the join graph K + K m ,obtained by joining each of two vertices to each vertex of the complete graph K m withedges, has property E ( m − , m -extendable.A surface is a connected compact Hausdorff space which is locally homeomorphic to anopen disc in the plane. If a surface Σ is obtained from the sphere by adding some number g ≥ g > orientable ofgenus g = g (Σ) (resp. non-orientable of genus ¯ g = ¯ g (Σ)). We shall follow the usual notationof the surface of orientable genus g (resp. non-orientable genus ¯ g ) by S g (resp. N ¯ g ).Let µ (Σ) be the smallest integer k such that no graphs embedded on the surface Σ are k -extendable. Dean [4] presented an elegant formula that µ (Σ) = 2 + ⌊ p − χ (Σ) ⌋ , (1)where χ (Σ) is the Euler characteristic of a surface Σ, i.e. χ (Σ) = 2 − g if Σ is an orientablesurface of genus g and χ (Σ) = 2 − ¯ g if Σ is a non-orientable surface of genus ¯ g . For thesurfaces Σ with small genus such as the sphere, the projective plane, the torus and the Kleinbottle, the following results show that no graphs embedded on Σ are E ( µ (Σ) − , emma 1.6. (i)([2]) No planar graph is E (2 , ;(ii) ([3])No projective planar graph is E (2 , ;(iii) ([3])If G is toroidal, then G is not E (3 , ;(iv) ([3])If G is embedded on the Klein bottle, then G is not E (3 , . In this paper we obtain the following general result, which will be proved in next section.
Theorem 1.7.
For any surface Σ , no graphs embedded on Σ are E ( µ (Σ) − , . Furthermore, we obtain that if a graph G embedded on a surface has enough manyvertices, then G has no property E ( k − ,
1) for each integer k ≥
4. Precisely, we have thefollowing result; its proof will be given in Section 3.
Theorem 1.8.
Let G be a graph with genus g (resp. non-orientable genus ¯ g ). Then if | V ( G ) | ≥ ⌊ g − k − ⌋ + 1 (resp. | V ( G ) | ≥ ⌊ g − k − ⌋ + 1 ), G is not E ( k − , for each integer k ≥ . Combining Theorem 1.8 with Lemma 1.5, we have an immediate consequence as follows.
Corollary 1.9. ([17]) Let G be any connected graph of genus g (resp. non-orientable genus¯g). Then if | V ( G ) | ≥ ⌊ g − k − ⌋ + 1 (resp. ⌊ g − k − ⌋ + 1 ) for any integer k ≥ , G is not k -extendable. In particular, if we put k = 4 in the corollary, we can obtain the following result whichis also a main theorem that Aldred et al. recently obtained. Corollary 1.10. ([1]) Let G be any connected graph of genus g (resp. non-orientable genus¯g). Then if | V ( G ) | ≥ g − (resp. g − ), G is not 4-extendable. For a graph G , the genus γ ( G ) (resp. non-orientable genus ¯ γ ( G )) of it is the minimumgenus (resp. non-orientable genus) of all orientable (resp. non-orientable) surfaces in which G can be embedded. An embedding ˜ G of a graph G on an orientable surface S k (resp. anon-orientable surface N k ) is said to be minimal if γ ( G ) = k (resp. ¯ γ ( G ) = k ) and ifeach component of Σ − ˜ G is homeomorphic to an open disc. Lemma 2.1. ([16]) Every minimal orientable embedding of a graph G is a -cell embedding. Lemma 2.2. ([7]) Every graph G has a minimal non-orientable embedding which is 2-cell. v be any vertex of a graph G embedded on an orientable surface of genus g (resp. anon-orientable surface of genus ¯ g ). Define the Euler contribution of the vertex v to be φ ( v ) = 1 − deg( v )2 + deg( v ) X i =1 f i , (2)where the sum runs over the face angles at vertex v , f i denotes the size of the ith face at v and deg( v ) denotes the degree of v . Lemma 2.3. ([5]) Let G be a connected graph 2-celluarlly embedded on some surface Σ oforientable genus g (resp. non-orientable genus ¯ g ). Then P v φ ( v ) = χ (Σ) . For a vertex v , it is called a control point if φ ( v ) ≥ χ (Σ) | V ( G ) | . If G is 2-cellularly embeddedon the surface Σ, then G must have at least one control point by Lemma 2.3.Let δ ( G ) denote the minimum degree of the vertices in G . The following lemma is asimple observation, which gives a lower bound of δ ( G ) of a graph G with E ( m, Lemma 2.4.
If a graph G is E ( m, for m ≥ , then δ ( G ) ≥ m + 2 .Proof. By Lemma 1.3, G is E ( m, δ ( G ) ≥ m + 1 by Lemma 1.1. Supposeto the contrary that there exists a vertex v with degree m + 1. Then G [ N ( v )] cannotcontain a matching of size 1 by Lemma 1.2; that is, N ( v ) is an independent set of G . Let N ( v ) = { v , v , ..., v m +1 } , V = { v , v , ..., v m } and R = V ( G ) \ N [ v ], where N [ v ] = N ( v ) ∪{ v } .Let G [ V, R ] be the induced bipartite graph of G with bipartition V and R . Then every vertexin V is adjacent to at least m vertices in R . Hence it can easily be seen that G [ V, R ] hasa matching M of size m saturating V . Let N = { vv m +1 } . Obviously, there is no perfectmatching F of G satisfying that M ⊆ F and N ∩ F = ∅ . This contradicts that G is E ( m, Proof of Theorem 1.7 . Since µ (Σ) increases as g (resp. ¯ g ) does and a graph embedded on asurface with small genus must be embedded on some surface with larger genus, it suffices toprove that any graph minimally embedded on the surface Σ is not E ( µ (Σ) − ,
1) by Lemma1.4. In the following, we may assume that G is minimally and 2-cell embedded on the surfaceΣ by Lemmas 2.1 and 2.2.By Lemma 1.6, the theorem holds for the surfaces S , S , N and N . Hereafter, we willrestrict our considerations on the other surfaces Σ. Consequently, χ (Σ) ≤ − µ (Σ) ≥ G is E ( µ (Σ) − , | V ( G ) | ≥ µ (Σ) + 1), and δ ( G ) ≥ µ (Σ) + 1 ≥ G is a 2-cell embedding on the surface Σ, it hasa control point v . Let y := deg( v ) and let x be the number of the triangular faces at v .4 laim 1. G is not E ( y − ⌈ x ⌉ , x = y and y is odd, then there is a matching of size ⌊ x ⌋ in G [ N ( v )]. Hence G is not E ( ⌊ x ⌋ , G is not E ( y − ⌈ x ⌉ , ⌈ x ⌉ in G [ N ( v )]. Then G is not ( y − ⌈ x ⌉ )-extendable by Lemma 1.2. Hence G is not E ( y − ⌈ x ⌉ , φ ( v ) ≥ χ (Σ) | V ( G ) | , we have y ≤ y X i =1 f i − χ (Σ) | V ( G ) | ≤ x y − x − χ (Σ)2( µ (Σ) + 1) , which implies that y ≤ x − χ (Σ) µ (Σ) + 1 . Let c := 4 − χ (Σ) µ (Σ) + 1 . (3)Then c > y − ⌈ x ⌉ ≤ y − x ≤ y − x ≤ c . Claim 2. G is not E ( ⌊ c ⌋ − , y − ⌈ x ⌉ ≤ y − x ≤ c −
1, then G is not E ( ⌊ c ⌋ − ,
1) by Lemma 1.4 and Claim 1.In what follows we suppose that y − x > c −
1. Combining this with y − x ≤ c , we havethat x ≤
5, and all possible cases of pairs of non-negative integers ( x, y ) are as follows:(0 , ⌊ c ⌋ ) , (1 , ⌊ c ⌋ ) , (1 , ⌊ c ⌋ + 1) , (2 , ⌊ c ⌋ + 1) , (3 , ⌊ c ⌋ + 1) , (4 , ⌊ c ⌋ + 2) , and (5 , ⌊ c ⌋ + 2).Suppose to the contrary that G is E ( ⌊ c ⌋ − , G is ( ⌊ c ⌋ − δ ( G ) ≥ ⌊ c ⌋ + 1 by Lemma 2.4. Hence the first two cases (0 , ⌊ c ⌋ ) and (1 , ⌊ c ⌋ )are impossible. If y = ⌊ c ⌋ + 1, since deg( v ) = y = ( ⌊ c ⌋ −
1) + 2, G [ N ( v )] cannot contain amatching of size 2 by Lemma 1.2.For convenience, let v , v , ..., v y be the vertices adjacent to v arranged clockwise at v in G . Similar to the notation in the proof of Lemma 2.4, let R = V ( G ) \ N [ v ] and G [ V, R ]denote the induced bipartite graph of G with bipartition V and R , where V ⊆ V ( G ) \ R .For ( x, y ) = (1 , ⌊ c ⌋ + 1), G [ N ( v )] cannot contain a matching of size 2. Assume that thetriangular face at v is vv v . Hence each v i , 3 ≤ i ≤ ⌊ c ⌋ + 1, can only be adjacent to v and v in N ( v ), and has at least ⌊ c ⌋ − R . Let V := { v , v , ..., v ⌊ c ⌋ } . Thenthere is a matching M ′ of size ⌊ c ⌋ − G [ V, R ]. Let M := M ′ ∪ { v v } and N := { vv ⌊ c ⌋ +1 } .Obviously, there is no perfect matching F satisfying that M ⊆ F and N ∩ F = ∅ , acontradiction.For ( x, y ) = (2 , ⌊ c ⌋ + 1), G [ N ( v )] cannot contain a matching of size 2, and the twotriangular faces at v must be adjacent. Hence we can assume that they are vv v and vv v .Each v i , 4 ≤ i ≤ ⌊ c ⌋ + 1, can only be adjacent to v in N ( v ), and has at least ⌊ c ⌋ − R . Let V := { v , v , ..., v ⌊ c ⌋ +1 } . Then we can find a matching M ′ of size ⌊ c ⌋ − G [ V, R ]. Let M = M ′ ∪ { v v } and N = { vv } . Consequently, it is impossible to find aperfect matching F satisfying that M ⊆ F and N ∩ F = ∅ , a contradiction.For ( x, y ) = (3 , ⌊ c ⌋ + 1), G [ N ( v )] contains a matching of size ⌈ ⌉ =2. This would beimpossible.If y = ⌊ c ⌋ + 2, since deg( v ) = ( ⌊ c ⌋ −
1) + 3, G [ N ( v )] cannot contain a matching of size 3by Lemma 1.2. Hence ( x, y ) = (5 , ⌊ c ⌋ + 2) would also be impossible since G [ N ( v )] containsa matching of size ⌈ ⌉ =3.For the remaining case ( x, y ) = (4 , ⌊ c ⌋ + 2), G [ N ( v )] cannot contain a matching of size3. Then the four triangular faces at v must have the following two cases. Case 1 . The fourtriangular faces are vv i v i +1 , 1 ≤ i ≤
4. Each v i , 6 ≤ i ≤ ⌊ c ⌋ + 2, can only be adjacentto v or v , and has at least ⌊ c ⌋ − R . Let V := { v , v , ..., v ⌊ c ⌋ +2 } .Then we can find a matching M ′ of size ⌊ c ⌋ − G [ V, R ]. Let M = M ′ ∪ { v v , v v } and N = { vv } . Obviously, there is no perfect matching F satisfying that M ⊆ F and N ∩ F = ∅ , a contradiction. Case 2 . The four triangular faces are vv i v i +1 for i = 1 , vv j v j +1 for j = t, t + 1, where t = 1 , , , ⌊ c ⌋ + 1 , ⌊ c ⌋ + 2. Then each v i , i = 1 , , , t, t + 1and t + 2, can only be adjacent to v and v t +1 , and has at least ⌊ c ⌋ − R . v can only be adjacent to v , v and v t +1 in G [ N ( v )], and has at least ⌊ c ⌋ − R . Let V := N ( v ) − { v , v , v t , v t +1 , v t +2 } . Then we can find a matching M ′ ofsize ⌊ c ⌋ − G [ V, R ]. Set M = M ′ ∪ { v v , v t v t +1 } and N = { vv t +2 } . Obviously, there isno perfect matching F satisfying that M ⊆ F and N ∩ F = ∅ , a contradiction. Hence theclaim holds. Claim 3. ⌊ c ⌋ ≤ µ (Σ).In fact, the inequality was stated in [4] without proof. Here we present a simple proof.Owing to the expressions (1) and (3) of µ (Σ) and c , it suffices to prove that ⌊ − χ (Σ)3 + ⌊ p − χ (Σ) ⌋ ⌋ ≤ ⌊ p − χ (Σ) ⌋ . Then we have the following implications: ⌊ − χ (Σ)3+ ⌊ √ − χ (Σ) ⌋ ⌋ ≤ ⌊ p − χ (Σ) ⌋⇐⇒ − χ (Σ)3+ ⌊ √ − χ (Σ) ⌋ < ⌊ p − χ (Σ) ⌋⇐⇒ ⌊ p − χ (Σ) ⌋ − χ (Σ) < ( ⌊ p − χ (Σ) ⌋ ) + 5 ⌊ p − χ (Σ) ⌋ + 6 ⇐⇒ − χ (Σ) < ( ⌊ p − χ (Σ) ⌋ + 1) Since the last inequality obviously holds, the claim follows.By the above arguments, we know that G is E ( µ (Σ) − ,
1) but not E ( ⌊ c ⌋ − , ⌊ c ⌋ − > µ (Σ) − (cid:3) Proof of Theorem 1.8
Suppose to the contrary that G is E ( k − , G is E ( k − ,
0) and δ ( G ) ≥ k + 1 byLemmas 1.3 and 2.4. We can assume that G is a 2-cell embedding on the surface S g (resp. N ¯ g ) by Lemmas 2.1 and 2.2. In the following, we mainly prove that φ ( v ) ≤ − k − for anyvertex v ∈ V ( G ). If it holds, by Lemma 2.3 we have χ (Σ) = P v φ ( v ) ≤ − k − | V ( G ) | , whichimplies that | V ( G ) | ≤ − χ (Σ) k − for k ≥
4. This contradiction to the condition establishes thetheorem.Let d = deg( v ) = k + m . Then m ≥
1. For convenience, we assume that v , v , ..., v d are the vertices adjacent to v arranged clockwise at v in G . There are three cases to beconsidered. Case 1. m ≥
3. Since d = ( k −
1) + m + 1 and G is ( k − G [ N ( v )] cannotcontain a matching of size m + 1 by Lemma 1.2. If there are at most 2 m triangular faces at v , then we have φ ( v ) ≤ − d m k + m − m − k − m + 1212 ≤ − k − − k . Otherwise, there are exactly 2 m + 1 triangular faces at v and d = 2 m + 1 = 2 k −
1. Let M := { v i v i +1 | ≤ i ≤ m − , i is odd } and N := { vv m +1 } . Then there exists no perfectmatching F such that M ⊆ F and N ∩ F = ∅ . But G is E ( k − , Case 2. m = 2. Since d = ( k −
1) + 3, G [ N ( v )] cannot contain a matching of size 3 byLemma 1.2. Hence there are at most four triangular face at v . It suffices to prove that thereare at most three triangular face at v . If so, we have that φ ( v ) ≤ − k +22 + + k − = − k .Suppose to the contrary that there are exactly four triangular faces at v . Then there aretwo cases to be considered. Subcase 2.1 . The four triangular faces are vv i v i +1 , where 1 ≤ i ≤
4. Then each v i , 6 ≤ i ≤ k + 2, can only be adjacent to v and v and has at least k + 1 − k − R , where R = V ( G ) \ N [ v ]. Let V := { v , v , ..., v k +2 } . Then we can find a matching M ′ of size k − G [ V, R ] of G . Let M := M ′ ∪ { v v , v v } and N := { vv } . Obviously, there is no perfect matching F satisfying that M ⊆ F and N ∩ F = ∅ , which contradicts the supposition that G has property E ( k − , Subcase 2.2 . The four triangular faces are vv i v i +1 , 1 ≤ i ≤
2, and vv j v j +1 , t ≤ j ≤ t + 1,where t = 1 , , , k + 1 , k + 2. Then for each v i , i = 1 , , , t, t + 1 and t + 2, it can only beadjacent to v and v t +1 , and has at least k − R . For the vertex v , itcan only be adjacent to v , v and v t +1 in N ( v ). Then it has at least k − R . Let V =: N ( v ) − { v , v , v t , v t +1 , v t +2 } . Then we can find a matching M ′ of size k − G [ V, R ] of G . Let M := M ′ ∪ { v v , v t v t +1 } and N := { vv t +2 } .7onsequently, it would be impossible to find a perfect matching F satisfying that M ⊆ F and N ∩ F = ∅ , a contradiction. Case 3. m = 1. Since d = ( k −
1) + 2, G [ N ( v )] cannot contain a matching of size2. Then there are at most two triangular faces at v . It suffices to prove that there are notriangular faces at v . If so, φ ( v ) = 1 − d + P di =1 1 f i ≤ − d + d = − k .If there is exactly one triangular face at v , suppose that it is vv v . Since G [ N ( v )] cannotcontain a matching of size 2, each vertex v i , where 3 ≤ i ≤ k , can only be adjacent to v and v in N ( v ). Consequently, it is adjacent to at least k − R . Let V = { v , v , ..., v k } .Then we can find a matching M ′ of size k − G [ V, R ] of G .Set M = M ′ ∪ { v v } and N = { vv k +1 } . Then there is no perfect matching F satisfyingthat M ⊆ F and N ∩ F = ∅ , a contradiction.Otherwise, there are exactly two triangular faces at v , which must be adjacent faces, say vv i v i +1 , where i = 1 ,
2. Then each vertex v i , where 4 ≤ i ≤ k + 1, can only be adjacent to v in N ( v ), and is adjacent to at least k − R . Let V := { v , v , ..., v k +1 } . Thenwe can find a matching M ′ of size k − G [ V, R ] of G . Set M := M ′ ∪ { v v } and N := { vv } . Then there is no perfect matching F satisfying that M ⊆ F and N ∩ F = ∅ , a contradiction. References [1] R.E.L. Aldred, Ken-ichi Kawarabayashi, M.D. Plummer, On the matching extendabilityof graphs in surfaces, J. Combin. Theory Ser. B 98 (2008) 105-115.[2] R.E.L. Aldred and M.D. Plummer, On restricted matching extension in planar graphs,in: 17th British Combin. Conference (Canterbury 1999), Discrete Math. 231 (2001)73-79.[3] R.E.L. Aldred and M.D. Plummer, Restricted matching in graphs of small genus, Dis-crete Math. 308 (2008) 5907-5921.[4] N. Dean, The matching extendability of surfaces, J. Combin. Theory Ser. B 54 (1992)133-141.[5] H. Lebesgue, Quelques cons´equences simples de la formule d’Euler, J. Math. 9 (1940)27-43.[6] A. McGregor-Macdonald, The E ( m, n ) property, M.S. Thesis, University of Otago, 2000.[7] T. Parsons, G. Pica, T. Pisanski and A. Ventre, Orientably simple graphs, Math. Slovaca37 (1987) 391-394. 88] M.D. Plummer, A theorem on matchings in the plane, in: L.D. Andersen, et al. (Eds.),Proceedings of a Conference in Memory of Gabriel Dirac, Ann. Diacrete Math. Vol. 41,North-Holland, Amsterdam, 1989, pp. 347-354.[9] M.D. Plummer, Extending matchings in graphs: a survey, Discrete Math. 127 (1994)277-292.[10] M.D. Plummer, Extending matchings in graphs: an update, Cong. Numer. 116 (1996)3-32.[11] M.D. Plummer, Matching extension and the genus of a graph, J. Combin. Theory Ser.B 44 (1986) 329-337.[12] M.D. Plummer, Matching extension in bipartite graphs, in: Proceedings of the Seven-teenth Southeastern Conference on Combinatorics, Graph Theory and Computing, in:Congr. Numer., LIV Utilitas Math., Winnipeg, 1986, pp. 245-258.[13] M.D. Plummer, On nn