Equivalence between Extendibility and Factor-Criticality
aa r X i v : . [ m a t h . C O ] N ov Equivalence between Extendibility andFactor-Criticality ∗ Zan-Bo Zhang † , Tao Wang , Dingjun Lou Department of Computer Science,Sun Yat-sen University, Guangzhou 510275, China Department of Computer Engineering, GuangdongIndustry Technical College, Guangzhou 510300, China Center for Combinatorics, LPMC,Nankai University, Tianjin 300071, China
Abstract
In this paper, we show that if k ≥ ( ν + 2) /
4, where ν denotes theorder of a graph, a non-bipartite graph G is k -extendable if and onlyif it is 2 k -factor-critical. If k ≥ ( ν − /
4, a graph G is k -extendableif and only if it is (2 k + 1)-factor-critical. We also give examples toshow that the two bounds are best possible. Our results are answersto a problem posted by Favaron [3] and Yu [11]. Key words : n -factor-critical, n -critical, k -extendable, k -extendable All graphs considered in this paper are finite, connected, undirected andsimple. Let G be a graph, vertex set and edge set of G are denoted by V ( G ) and E ( G ). Let S ⊆ V ( G ), we use G [ S ] to denote the subgraph of G induced by S and G − S to denote the subgraph G [ V ( G ) \ S ]. Let G and G be two disjoint graphs. The union G ∪ G is the graph with vertexset V ( G ) ∪ V ( G ) and edge set E ( G ) ∪ E ( G ). The join G ∨ G is thegraph obtained from G ∪ G by joining each vertex of G to each vertexof G . The complete graph on n vertices and its complement are denoted ∗ Work supported by the Scientific Research Foundation of Guangdong Industry Tech-nical College, granted No. 2005-11. † Corresponding Author. Email address: [email protected]. y K n and I n . Let X and Y be two disjoint subsets of V ( G ), the numberof edges of G from X to Y is denoted by e ( X, Y ). For other terminologiesand notations not defined in this paper, we refer the readers to [1].A matching M of G is a subset of E ( G ) in which no two edges havea common end-vertex. M is said to be a perfect matching if it covers allvertices of G . A graph G is said to be k -extendable for 0 ≤ k ≤ ( ν − / k and any matching in G ofsize k is contained in a perfect matching of G . G is said to be minimal k -extendable if G is k -extendable and G − e is not k -extendable for each e ∈ E ( G ). The concept of k -extendable graphs was introduced by Plummerin [8]. In [10], Yu generalized the idea of k -extendibility to k -extendibilityfor graph of odd order. A graph G is said to be k -extendable if (1) for anyvertex v of G there exists a matching of size k in G − v , and (2) for everyvertex v of G , every matching of size k in G − v is contained in a perfectmatching of G − v .A graph G is said to be n -factor-critical , or n -critical , for 0 ≤ n ≤ ν − G − S has a perfect matching for any S ⊆ V ( G ) with | S | = n . For n = 1,2, that is factor-critical and bicritical . G is called minimal n -factor-critical if G is n -factor-critical but G − e is not n -factor-critical for any e ∈ E ( G ).The concept of n -factor-critical graphs was introduced by Favaron [2] andYu [10], independently.It is easy to verify the following theorem. Theorem 1.1.
For ≤ k ≤ ν/ − , a k -factor-critical graph is k -extendable, and a (2 k + 1) -factor-critical graph is k -extendable. The reverse of Theorem 1.1 does not hold in general. For example, a k -extendable bipartite graphs can not be n -factor-critical for any n > k -extendable non-bipartite graphs and n -factor-critical graphs. Most of theresults can be viewed as answers to the following problem, which has beenposted by Favaron [3] and Yu [11], in slightly different forms. Problem 1.
Does there exist a non-null function f ( k ) such that every k -extendable non-bipartite graph of even order ν ≥ k + 2 is f ( k ) -factor-critical? The following two results of Plummer [8] are answers to k = 2, 3. Theorem 1.2.
Let G be 2-extendable and non-bipartite with ν ≥ , then G is bicritical. Theorem 1.3.
Let G be 3-extendable and bicritical with ν ≥ , then G − e is again bicritical for any e ∈ E ( G ) . And they have been generalized for all k ≥
0, as below. heorem 1.4. ( Favaron [3], Liu and Yu [5] ) For even integer k ≥ ,every connected, non-bipartite, k -extendable graph of even order ν > k is k -factor-critical. Theorem 1.5. ( Favaron [3] ) For even integer k ≥ , every connectednon-bipartite, ( k + 1) -extendable graph G of even order ν ≥ k + 4 is k -factor-critical. Moreover, G − e is k -factor-critical for every edge e of G . In light of Theorem 1.1, if under some conditions k -extendable graphs are2 k -factor-critical, then the two classes of graphs are equal. The followingresults show that this happens when k is large relative to ν . Theorem 1.6. ( Favaron and Shi [4] ) Every (( ν/ − -extendable non-bipartite graph with ν ≥ is ( ν − -factor-critical. Theorem 1.7. ( Yu [11] ) Let G be a non-bipartite graph of even order and k an integer. If G is k -extendable and k ≥ ν + 1) / , then G is k -factor-critical. Following this direction, we give a better lower bound of k and show thatit is the best possible. Furthermore, we show a similar equivalent relation-ship between (2 k + 1)-factor-critical graphs and k -extendable graphs.The following lemmas will be used in the proofs of the main results. Lemma 1.8. ( Plummer [8] ) If G is a k -extendable graph on ν ≥ k + 2 vertices where k ≥ , then G is also ( k − -extendable. Lemma 1.9. ( Yu [10] ) If G is a k -extendable graph, then G is also ( k − -extendable. Lemma 1.10. ( Lou and Yu [7] ) If G is a k -extendable graph with k ≥ ν/ ,then either G is bipartite or κ ( G ) ≥ k . Lemma 1.11. If G is a k -extendable graph, then G is also m -extendablefor all integers ≤ m ≤ k . If G is a k -extendable graph, then G is also m -extendable for all integers ≤ m ≤ k .Proof. Apply repeatedly Lemma 1.8 and Lemma 1.9.
Theorem 2.1. If k ≥ ( ν + 2) / , then a non-bipartite graph G is k -extendable if and only if it is k -factor-critical .roof. We only need to prove that if k ≥ ( ν + 2) /
4, a k -extendable non-bipartite graph is 2 k -factor-critical.Let G be a k -extendable non-bipartite graph satisfying k ≥ ( ν + 2) / k -factor-critical. Then, there exists a vertex set S ⊆ V ( G ) withorder 2 k , such that G − S has no perfect matching. Moreover, we choose S so that the size of the maximum matching of G [ S ] has the maximum value r . Clearly, r ≤ k − M S be a maximum matching of G [ S ], then there exists two vertices u and u in G [ S ] that are not covered by M S . By Lemma 1.11, M S is contained in a perfect matching M of G . Let u i v i ∈ M , where v i ∈ V ( G − S ), i = 1, 2. Let S ′ = ( S \{ u ) } ∪ { v } . Then M S ∪ { u v } is amatching of G [ S ′ ] of size r + 1. By the choice of S , G − S ′ has a perfectmatching M ¯ S ′ , and | M ¯ S ′ | ≤ k −
1. By Lemma 1.11, M ¯ S ′ is contained ina perfect matching M ′ of G . Clearly, M ′ ∩ E ( G [ S ′ ]) is a perfect matchingof G [ S ′ ] and M ′ ∩ E ( G [ S ]) is a matching of G [ S ] of size k −
1. Therefore, r = k −
1. Then M ¯ S = M ∩ E ( G − S ) is a maximum matching of G − S of size r = | V ( G − S ) | / − ≤ k −
2, and v v / ∈ E ( G ).Let M S = { x x , . . . , x k − x k − } , M ¯ S = { y y , . . . , y r − y r } . If v y , v y ∈ E ( G ), then M ¯ S ∪ { v y , v y }\ y y is a perfect matching of G − S ,contradicting our assumption. Hence |{ v y , v y } ∩ E ( G ) | ≤
1. Simi-larly, |{ v y , v y } ∩ E ( G ) | ≤
1. So e ( { v , v } , { y , y } ) ≤
2. Similarly, e ( { v , v } , { y i − , y i } ) ≤ ≤ i ≤ r .If v x , v x ∈ E ( G ), then M ¯ S ∪ { v x , v x } is a matching of G ofsize no more than k . By Lemma 1.11, M ¯ S ∪ { v x , v x } is containedin a perfect matching M ′′ of G . But then ( M ′′ ∩ E ( G [ S ])) ∪ { x x } is aperfect matching of G [ S ], a contradiction. Hence, |{ v x , v x }∩ E ( G ) | ≤ |{ v x , v x }∩ E ( G ) | ≤
1. So e ( { v , v } , { x , x } ) ≤
2. Similarly, e ( { v , v } , { x i − , x i } ) ≤ ≤ i ≤ k − d ( v ) + d ( v ) ≤ k −
1) + 2 r + 4 ≤ k −
1) + 2( k −
2) + 4 = 4 k − . But by Lemma 1.10, δ ( G ) ≥ κ ( G ) ≥ k . So d ( v )+ d ( v ) ≥ k +2 k = 4 k ,a contradiction.To show that the lower bound in Theorem 2.1 is best possible, we considerthe following class of graphs. Let G ( k ) = ( K k − ∪ K ) ∨ ( K k − ∪ K ), k ≥
2. Then ν ( G ( k ) ) = 4 k and G ( k ) is non-bipartite. Theorem 2.2. G ( k ) is k -extendable but not k -factor-critical.Proof. Let G and G be two copies of K k − ∪ K and G ( k ) = G ∨ G . G ( k ) is not 2 k -factor-critical, since G = G ( k ) − V ( G ) does not have aperfect matching. Now we prove that G ( k ) is k -extendable.et M be a matching of size k in G , we show that G ( k ) − V ( M ) has aperfect matching. Let | M ∩ E ( G i ) | = k i , i = 1 ,
2, then k , k ≤ k − k ≥ k . The size of the maximummatching in G − V ( M ) is no less than ⌊ (2 k − − ( k − k − k ) − k ) / ⌋ = ⌊ (( k −
1) + ( k + k ) − k ) / ⌋ ≥ ⌊ (2 k − k ) / ⌋ = k − k . Therefore wecan find a matching M ′ of size k − k in G − V ( M ).In G ( k ) − V ( M ) − V ( M ′ ), half of the vertices are from G and the otherhalf are from G , hence we can find nonadjacent edges from G to G covering all vertices in it. So we get a perfect matching in G ( k ) − V ( M )and G ( k ) is k -extendable.Now we divert our attention to k -extendable graphs and (2 k +1)-factor-critical graphs. Note that by definition a k -extendable graph can neverbe bipartite. Theorem 2.3. If k ≥ ( ν − / , then a graph G is k -extendable if andonly if it is (2 k + 1) -factor-critical.Proof. We only need to prove that for k ≥ ( ν − /
4, a k -extendable graph G is (2 k + 1)-factor-critical.Suppose that G is a k -extendable graph with k ≥ ( ν − /
4, but not(2 k + 1)-factor-critical. Then, there exists a set S ⊆ V ( G ) of order 2 k + 1,such that there is no perfect matching in G − S . Denote by r the size ofthe maximum matching in G [ S ]. Clearly, r ≤ k − M S be a maximum matching of G [ S ], and v be a vertex of G [ S ]not covered by M S . Then by Lemma 1.11, M S is contained in a perfectmatching M of G − v . Then M ∩ E ( G − S ) is a matching of G − S of sizeat most | V ( G − S ) | / − ≤ k .Let v be a vertex in G − S not covered by M ∩ E ( G − S ), then M ∩ E ( G − S )is contained in a perfect matching M ′ of G − v . But M ′ ∩ E ( G [ S ]) is amatching of G [ S ] of size at least r + 1, a contradiction.We present a class of graphs below to show that the bound in Theorem2.3 is best possible. Let H ( k ) = I k +2 ∨ ( K k +3 ∪ K k ). Then ν ( H ( k ) ) = 4 k +5. Theorem 2.4. H ( k ) is k -extendable but not (2 k + 1) -factor-critical.Proof. Let H = I k +2 , H = K k +3 , H = K k and H ( k ) = H ∨ ( H ∪ H ).Let S be a subset of V ( H ) of order k − u ∈ V ( H ). Let S = V ( H ) ∪ S ∪ { u } . Then | S | = 2 k + 1 and H ( k ) − S does not have a perfectmatching. Therefore H ( k ) is not (2 k + 1)-factor-critical.To prove the k -extendibility of H ( k ) , we let v ∈ V ( H ( k ) ), M be amatching of size k in H ( k ) − v and S = { v } ∪ V ( M ). We show that H ( k ) − S has a perfect matching.et V = V ( H ) − S , V = V ( H ) − S and V = V ( H ) − S . The existenceof a perfect matching in H ( k ) − S is equivalent to the existence of a partitionof V into two subsets V ′ and V ′′ , such that | V ′ | ≤ | V | , | V ′′ | ≤ | V | , | V ′ | ≡ | V | (mod 2) and | V ′′ | ≡ | V | (mod 2). Since | V | + | V | + | V | = | V ( G ) | − (2 k + 1) is even, | V | and | V | + | V | have the same parity. Andsince | V | + | V | ≥ ( k + 3) + 2 k − (2 k + 1) = k + 2 ≥ | V | ≥ k + 2 − − k = 1,such a partition can always be obtained. Hence we find a perfect matchingin H ( k ) − S and H ( k ) is k -extendable. As we have pointed out earlier, a k -extendable bipartite graph G can notbe n -factor-critical for any n >
0. This is because we can choose a vertexset S of order n so that G − S is not balanced. However, for n = 2 k , if wekeep the two partitions of G − S balanced when we choose S , then G − S does have a perfect matching. This is a result by Plummer [9]. Theorem 3.1.
Let G be a connected bipartite graph with bipartition (U,W)and suppose k is a positive integer such that k ≤ ν/ − . Then G is k -extendable if and only if for all u , . . . , u k ∈ U and w , . . . , w k ∈ W , G ′ = G − u − · · · − u k − w − · · · − w k has a perfect matching. Hence, following the terms in the definition of n -factor-critical graphs,if we define “2 k -factor-criticality” in a balanced bipartite graph G so thatwe keep the two partitions of G − S balanced when choosing S , then G is k -extendable if and only if it is “2 k -factor-critical”, for 0 ≤ k ≤ ν/ − κ ( G ) ≥ k + 1 for a k -extendable graph G .Hence δ ( G ) ≥ κ ( G ) ≥ k + 1. For minimal k -extendable bipartite graphs,the following result of Lou [6] shows that the bound can always be reached. Theorem 3.2.
Every minimal k -extendable bipartite graph G with bipar-tition (U,W) has at least k + 2 vertices of degree k + 1 . Furthermore, both U and W contain at least k + 1 vertices of degree k + 1 . While for minimal k -extendable non-bipartite graphs we have not foundsuch a simple characterization. When k = 1, the minimum degree can be2 or 3. And no result is known for k ≥
2. Illuminated by Lemma 1.10, Louand Yu [7] raised the following conjecture.
Conjecture 1.
Let G be a minimal k -extendable graph on ν vertices with ν/ ≤ k + 1 . Then δ ( G ) = k + 1 , k or k + 1 . For minimal n -factor-critical graphs, Favaron and Shi [4] raised the fol-lowing conjecture. onjecture 2. Every minimal n -factor-critical graph G has δ ( G ) = n + 1 . By the results obtained, we see that except the case that ν = 4 k , Con-jecture 1 is actually part of Conjecture 2 and the value 2 k in Conjecture 1can be excluded. Acknowledgements
We thank Professor Qinglin Yu for suggesting the problem and his valu-able advices.
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