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Sufficient conditions for triangle-free graphs to be super k-restricted edge-connected
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Sufficient conditions for triangle-free graphs to be super k-restricted edge-connected ✩ Jun Yuan a , Aixia Liu b,∗ a b
School of Applied Science, Taiyuan University of Science and Technology, Taiyuan, Shanxi 030024, People’s Republic of China School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 03006, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 11 June 2013 Received in revised form 7 April 2015 Accepted 3 September 2015 Available online xxxx Communicated by Tsan-sheng Hsu Keywords: Fault tolerance Restricted edge connectivity Super restricted edge connected connectivity Triangle-free graph
a b s t r a c t An edge cut S of a connected graph G = ( V , E ) is a k-restricted edge cut if every component of G − S contains at least k vertices. A graph is said to be super k-restricted edge-connected if every minimum k-restricted edge cut is a set of edges incident to a certain connected subgraph of order k. Let k be a positive integer, and let G be a connected triangle-free graph of order n ≥ 2k. In this paper, we prove that if the minimum degree
δ(G ) ≥ k + 1 − (−1)k and there are at least k +
common vertices in the neighbor sets of each pair of nonadjacent vertices in G, then G is super k-restricted edge-connected. © 2015 Elsevier B.V. All rights reserved.
1. Terminology and introduction For graph-theoretical terminology and notation not defined here, we follow [2]. In this paper, we consider only finite, undirected, and simple graphs G with the vertex set V (G ) and the edge set E (G ). For two vertices u , v ∈ V (G ), the distance between u and v is the length of the shortest path from u to v in G. The diameter of G is the maximum distance between two vertices of G. For two disjoint vertex set U 1 , U 2 ⊂ V (G ), the set of edges with one end in U 1 and the other in U 2 is denoted by [U 1 , U 2 ]. If U 1 = {u }, then [U 1 , U 2 ] is abbreviated as [u , U 2 ]. We use G [U ] to denote the subgraph of G induced by U ⊆ V (G ), and U = V (G ) \ U the complement of U . Let U be a vertex subset of V (G ). Define the open neighborhood N G (U ) of U in G as the set { v ∈ U : |[ v , U ]| ≥ 1}. In particular, if U = { v }, then N G (U ) is denoted by N G ( v ). The degree ✩ This work is supported by the National Science Foundation of China (61402317, 61303020) and the Natural Science Foundation of Shanxi Province (2012021001-2). Corresponding author. E-mail address: [email protected] (A. Liu).
*
http://dx.doi.org/10.1016/j.ipl.2015.09.005 0020-0190/© 2015 Elsevier B.V. All rights reserved.
1+(−1)k 2
d G ( v ) of a vertex v is the number of vertices in N G ( v ). Let δ(G ) denote the minimum vertex degree in G. For N G (U ) and d G ( v ), we usually omit the subscript for the graph when no confusion arises. An edge subset M of E (G ) is called a matching in G if no two edges of M are adjacent in G. M is a maximum matching if G has no matching M with | M | > | M |. A subset S of V (G ) is called an independent set of G if no two vertices of S are adjacent in G. An independent set S is maximum if G has no independent set S with | S | > | S |. The number of vertices in a maximum independent set of G is called the independence number of G and is denoted by α (G ). A subset K of V (G ) such that every edge of G has at least one end in K is called a covering of G. A covering K is minimum if G has no covering K with | K | < | K |. The number of vertices in a minimum covering of G is called the covering number of G and is denoted by β(G ). A network can be conveniently modeled as a graph. One fundamental consideration in the design of networks is reliability. A classic measure of the reliability of a network is the edge connectivity λ(G ). In general, the larger λ(G ) is, the more reliable the network is. A more refined measure known as restricted edge connectivity was pro-
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posed by Esfahanian and Hakimi [4], which was further generalized to k-restricted edge connectivity by Fàbrega and Fiol [5] (called k-extra edge connectivity in their paper). For a connected graph G, an edge set S ⊂ E (G ) is said to be a k-restricted edge cut if G − S is disconnected and every component of G − S contains at least k vertices. The cardinality of the minimum k-restricted edge cut of G, denoted as λk (G ), is called the k-restricted edge connectivity of G. It should be emphasized that not all connected graphs have k-restricted edge cuts. A connected graph G is said to be λk -connected if G has a k-restricted edge cut. Sufficient conditions for graphs to be λk -connected were shown by several authors [3,4,7,15]. According to recent studies on the k-restricted edge connectivity of G, it seems that the larger λk (G ) is, the more reliable the network is. So we expect λk (G ) to be as large as possible. Define ξk (G ) = min{|[U , U ]| : U ⊂ V (G ), |U | = k ≥ 1 and G [U ] is connected}. It has been shown that λk ≤ ξk holds for many graphs [3,7,15]. A connected graph G with λk (G ) = ξk (G ) is said to be λk -optimal. Furthermore, if every minimum k-restricted edge cut is a set of edges incident to a certain connected subgraph of order k, then G is said to be super k-restricted edge-connected, in short, super-λk . In particular, super-λ1 and super-λ2 are denoted as super-λ and super-λ , respectively. By definition, if G is super-λk , then G is λk -optimal. However, the converse is not true. For example, a cycle of length n ≥ 2k + 2 is λk -optimal, but not super-λk . Investigations on the restricted edge connectivity of graphs were made by several authors, for example, by [8,9,11–14,16]. In 2009, Wang and Lin [10] gave a neighborhood condition for graphs with diameter 2 to be super-λk graphs. Theorem 1.1. (See [10].) Let k be a positive integer and let G be a graph with order at least 2k. If | N (u ) ∩ N ( v )| ≥ 2k + 1 for all pairs u, v of nonadjacent vertices, then G is super-λk . For k = 2, 3, Shang and Zhang [8,9] once showed the following neighborhood conditions for graphs to be super-λk . Theorem 1.2. (See [8].) Let G be a connected triangle-free graph with order ν ≥ 4 and different from K 2,ν −2 . If | N (u ) ∩ N ( v )| ≥ 2 for all pairs u, v of nonadjacent vertices, then G is super-λ2 . Theorem 1.3. (See [9].) Let G be a triangle-free graph with order
ν ≥ 6 and minimum degree δ(G ) ≥ 5. If | N (u ) ∩ N ( v )| ≥ 2 for all pairs u , v of nonadjacent vertices, then G is super-λ3 . By contrast Theorems 1.2, 1.3 with Theorem 1.1, it is easy to see that the requirement of Theorem 1.1 is too high to determine a triangle-free graph to be super-λk for a genereal integer k. However, for a general integer k, the neighborhood condition for triangle-free graphs to be super-λk have not been given out yet. In this paper, we show a neighborhood condition for triangle-free graphs to be super-λk for general integer k. 2. Main result A vertex set U ⊂ V (G ) is called a λk -fragment if [U , U ] is a k-restricted edge cut of G with |[U , U ]| = λk (G ). For
a λk -fragment U , it is easy to see that G [U ] and G [U ] are both connected. In [13], the authors have proved that: Lemma 2.1. (See [13].) Let G be a λk -connected graph with λk (G ) ≤ ξk (G ) and let U be a λk -fragment of G. There exists no connected subgraph H of order k in G [U ] such that
|[ V ( H ), U \ V ( H )]| < |[U \ V ( H ), U ]|. Lemma 2.2. Let G be a λk -optimal triangle-free graph with minimum degree δ(G ) ≥ k + 1 − (−1)k . If G is not super-λk , then for any λk -fragment U of G with |U | ≥ k + 1 and |U | ≥ k + 1, the set
k U ∗ = { v ∈ U : |[ v , U ]| ≤ } = ∅. 2 Proof. Note that |[ v , U ]| is integer. It is sufficient to proof that U ∗ = { v ∈ U : |[ v , U ]| ≤ 2k } = ∅. Suppose on the contrary that U ∗ = { v ∈ U : |[ v , U ]| ≤ 2k } = ∅, that is
|[ v , U ]| ≥
k+1
(2.1)
2
for any v ∈ U . Let W be a vertex set of k vertices in U satisfying the following conditions: (i) G [ W ] is connected; (ii) f ( W ) = max{|[ v , W ]| : v ∈ U \ W } is as small as possible; denote the number by f 0 ; (iii) g ( W ) = |{ v : v ∈ U \ W , |[ v , W ]| = f 0 }| is as small as possible; denote the number by g 0 . By (2.1),
|[U \ W , U ]| =
|[ v , U ]| ≥
k+1
v ∈U \ W
2
|U \ W |.
(2.2)
Consider the following two cases. Case 1. f 0 ≤ 2k .
In this case, for any v ∈ U \ W , we have that |[ v , W ]| ≤ 2k .
This implies that |[ W ,U \ W ]| =
v ∈U \ W
|[ v , W ]| ≤ 2k |U \ W |.
Combining this with (2.2), we have that |[ W , U \ W ]| <
|[U \ W , U ]|, contradicting to Lemma 2.1. Case 2. f 0 > 2k .
1 We first show f 0 ≥ 2k + 1. Suppose that f 0 = k+ . By 2 the definition of f 0 , we have that
|[ W , U \ W ]| ≤ f 0 |U \ W | =
k+1 2
|U \ W |.
(2.3)
Combining this with (2.2), we have |[ W , U \ W ]| ≤ |[U \ W , U ]|. By Lemma 2.1, |[ W , U \ W ]| = |[U \ W , U ]|. This implies that the inequalities of (2.2) and (2.3) must be 1 equalities, that is |[ W , U \ W ]| = k+ |U \ W | = |[U \ W , U ]|. 2 1 It follows that |[ v , W ]| = |[ v , U ]| = k+ for any v ∈ U \ W . 2
Combining this with δ(G ) ≥ k + 1 − (−1)k , there exist at least one edge in G [U \ W ], say xy. Since G is triangle-free, it follows that N (x) ∩ N ( y ) = ∅. So d G [ W ] (x) + d G [ W ] ( y ) ≤ k. 1 On the other hand, by |[ v , W ]| = k+ for any v ∈ U \ W , 2
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we can conclude that d G [ W ] (x) + d G [ W ] ( y ) = k + 1, a contradiction. Therefore,
f0 ≥
k 2
+ 1.
(2.4)
Next, we shall construct a new connected subgraph W of order k such that f ( W ) < f ( W ) or f ( W ) = f ( W ) and g ( W ) < g ( w ). Let u be a vertex in U \ W such that |[u , W ]| = f 0 , that is | N (u ) ∩ W | = f 0 Since G is trianglefree, N (u ) ∩ W is an independent set of G. Let T be a spanning tree of G [ W ], and let S = { w : w ∈ N (u ) ∩ W , d T ( w ) = 1}. Clearly, the edge number ε ( T ) = k − 1 ≥ 2(| N (u ) ∩ W | − | S |) + | S | = 2 f 0 − | S | > k − | S |, and hence | S | > 1. Let w be a vertex of S. Then construct a new subgraph W such that W = W − { w } + {u }. Therefore, G [ W ] is connected and | W | = k. Since G is trianglefree, N ( w ) ∩ N (u ) = ∅. It follows that | N ( w ) ∩ W | ≤ | W | − | N (u ) ∩ W | = k − f 0 < k − 2k = 2k . Note that | N ( w ) ∩ W | is an integer. By the choice of w, we have that
k
| N ( w ) ∩ W | = | N ( w ) ∩ W | + 1 ≤ . 2
(2.5)
| N (u ) ∩ N ( v )| ≥ k for each pair of nonadjacent vertices u , v ∈ V (G ), then G is λk -optimal. Theorem 2.6. (See [1, Theorem 5.3, p. 74].) In a bipartite graph, the number of edges in a maximum matching is equal to the covering number. Theorem 2.7. (See [1, Corollary 7.1, p. 101].) Let G be a graph of order n, and let α (G ) and β(G ) be the independence number and the covering number. Then α (G ) + β(G ) = n. Theorem 2.8. Let k be a positive integer, and let G be a trianglefree graph of order n ≥ 2k and minimum degree δ(G ) ≥ k + 1 − (−1)k . If
| N (u ) ∩ N ( v )| ≥ k +
1 + (−1)k 2
(2.6)
Let v be a vertex in U \ ( W ∪ {u }). If uv ∈ E (G ), then, since G is triangle-free, we have that N (u ) ∩ N ( v ) = ∅. This implies that v is not adjacent to w and | N ( v ) ∩ W | ≤ | W | − | N (u ) ∩ W | = k − f 0 < 2k . Combining this with (2.4), we have that
|[ v , W ]| = |[ v , W ]| + 1 = |N (v ) ∩ W | + 1 <
k 2
+ 1 ≤ f 0.
(2.7)
If uv ∈ / E (G ), then
|[ v , W ]| ≤ |[ v , W ]|.
(2.8)
Combining (2.6), (2.7), and (2.8) with the definition of f 0 , we can conclude that 1, then f ( W )
(a) if g 0 = < f 0 , which contradicts condition (ii); (b) if g 0 ≥ 2, then f ( W ) = f 0 and g ( W ) = g 0 − 1, which contradicts condition (iii). The proof is completed.
Theorem 2.5. (See [14].) Let k be a positive integer, and let G be a triangle-free graph of order n ≥ 2k. If
for each pair of nonadjacent vertices u , v ∈ V (G ), then G is super-λk .
Combining (2.4) and (2.5), we have that
|N ( w ) ∩ W | < f 0 .
3
2
The following results play important roles in the proof of Theorem 2.8.
1+(−1)k
Proof. By the assumption that | N (u ) ∩ N ( v )| ≥ k + 2 for each pair of nonadjacent vertices u , v ∈ V (G ), the diameter of G is 2. If k = 1, then by δ(G ) ≥ k + 1 − (−1)k = 3 and Theorem 2.3, G is super-λ1 . If k = 2, then by assump1+(−1)k
= 3 for each tion, we have | N (u ) ∩ N ( v )| ≥ k + 2 pair of nonadjacent vertices u , v ∈ V (G ). It follows that G = K 2,n−2 . By Theorem 2.4, G is super-λ2 . Suppose k ≥ 3. By Theorem 2.5, G is λk -optimal and hence λk (G ) = ξk (G ). Suppose to the contrary that G is not super-λk . Then there exists a λk -fragment U such that |U | ≥ k + 1 and |U | ≥ k + 1. By Lemma 2.2, the set U ∗ = { v ∈ U : |[ v , U ]| ≤ 2k } = ∅, U ∗ = { v ∈ U : |[ v , U ]| ≤ 2k } = ∅. Let u be an arbitrary
element in U ∗ and v an arbitrary element in U ∗ . Claim 1. u is adjacent to v. Otherwise, by assumption, we have that
k+
1 + (−1)k 2
≤ | N (u ) ∩ N ( v )| = | N (u ) ∩ N ( v ) ∩ U | + | N (u ) ∩ N ( v ) ∩ U | ≤ | N ( v ) ∩ U | + | N (u ) ∩ U | k
≤ 2 , 2
Theorem 2.3. (See [6].) Let G be a graph of order n and minimum degree δ . Then G is super-λ if the diameter of G is 2 and G contains no the complete graph K δ with all its vertices of degree δ . Theorem 2.4. (See [8].) Let G be a connected triangle-free graph of order n ≥ 4 and different from K 2,n−2. If
| N (u ) ∩ N ( v )| ≥ 2 for all pairs u, v of nonadjacent vertices, then G is super-λ .
a contradiction. The proof of Claim 1 is completed. Claim 2. |U ∗ | ≤ 2k . By Claim 1, we have that U ∗ ⊆ N ( v ). It follows that U ∗ ⊆ N ( v ) ∩ U , and hence |U ∗ | ≤ | N ( v ) ∩ U | ≤ 2k . The proof of Claim 2 is completed. Combining Claim 1 with the fact that G is triangle-free, it can be deduced that U ∗ is an independent set of G. We consider the following two cases.
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4
Case 1. There is a vertex u 1 ∈ U ∗ such that |U \ ( N (u 1 ) ∩ U )| ≥ 2k . Since U ∗ is an independent set of G, we can deduce that U ∗ ⊆ U \ ( N (u 1 ) ∩ U ). By |U \ ( N (u 1 ) ∩ U )| ≥ 2k and Claim 2, there exists a set U 0 ⊆ U \ ( N (u 1 ) ∩ U ) such that U ∗ ⊆ U 0 and |U 0 | = 2k . Let U 0 = {u 1 , u 2 , · · · , u k }. 2
Clearly, u 1 is not adjacent to any vertex in U 0 \ {u 1 }. By assumption, we have that | N (u 1 ) ∩ N (u i ) ∩ U | = | N (u 1 ) ∩ 1) N (u i )| − | N (u 1 ) ∩ N (u i ) ∩ U | ≥ k + 1+(− − | N (u 1 ) ∩ U | ≥ 2 k
for i = 2, · · · , 2k . {v 1, v 2, · · · , v k } 2
k 2
Therefore, there exists a vertex set U 1 = such that v i ∈ N (u 1 ) ∩ N (u i +1 ) ∩ U for
i = 1, · · · , 2k − 1 and v k , v k ∈ N (u 1 ) ∩ N (u k ) ∩ U . 2 2 2 Let W = U 0 ∪ U 1 . Clearly, G [ W ] is connected. We shall show that the independent number α (G [ W ]) ≤ 2k . Let H = G [ W ] − E (G [U 0 ]). Then H is a bipartite graph with bipartition (U 0 , U 1 ). Clearly, the edge set {u i v i −1 : i = 2, · · · , 2k } ∪ {u 1 v k } is a maximum match2
ing of H (define {u i v i −1 : i = 2, · · · , 2k } = ∅ when k = 3). By Theorem 2.6 and Theorem 2.7, we have that α ( H ) = | V ( H )| − 2k = 2k . It follows that
k
α (G [ W ]) ≤ α ( H ) = .
(2.9)
2
Let v be an arbitrary vertex of U \ W . Since G is triangle-free, we have that N ( v ) ∩ W is an independent set of G. By (2.9), we have that | N ( v ) ∩ W | ≤ 2k . It follows
| N ( v ) ∩ W | ≤ 2k |U \ W |. On the other hand, since U ∗ ⊆ U 0 , we have that | N ( v ) ∩ U | ≥ k+2 1 for any v ∈ U \ W . It follows that |[U \ W , U ]| = k+1 v ∈U \ W | N ( v ) ∩ U | ≥ 2 |U \ W |. So, that |[ W , U \ W ]| =
v ∈U \ W
k
|[ W , U \ W ]| ≤ |U \ W | ≤ 2 ≤ |[U \ W , U ]|.
k+1 2
|U \ W |
If k is even, then |[ W , U \ W ]| < |[U \ W , U ]|. On the other hand, since λk (G ) = ξk (G ), by Lemma 2.1, we have that |[ W , U \ W ]| ≥ |[U \ W , U ]|, a contradiction. So, suppose k is odd. Then we have |[ W , U \ W ]| ≤ |[U \ W , U ]|. Since λk (G ) = ξk (G ), by Lemma 2.1, we have that |[ W , U \ W ]| ≥ |[U \ W , U ]| and hence |[ W , U \ W ]| = |[U \ W , U ]|. It im1 plies that | N ( v ) ∩ W | = | N ( v ) ∩ U | = k+ for any v ∈ U \ W . 2 So, we deduce that | N ( v ) ∩ N ( w ) ∩ W | ≥ | N ( v ) ∩ W | + | N ( w ) ∩ W | − | W | = 1 for any v , w ∈ U \ W . Note G is triangle-free. We have U \ W is an independent set and hence for any v ∈ U \ W , d( v ) = | N ( v ) ∩ W | + | N ( v ) ∩ U | = k + 1. It implies that δ(G ) ≤ k + 1. On the other hand, since k is odd, by assumption, δ(G ) ≥ k + 1 − (−1)k = k + 2, a contradiction. Case 2. |U \ ( N (u ) ∩ U )| < 2k for any u ∈ U ∗ . Let u be an arbitrary vertex of U ∗ , and let U 0 = U \ ( N (u ) ∩ U ). Note that U ∗ is an independent set of G and u ∈ U ∗ . By the definition of U 0 , U ∗ ⊆ U 0 . Let U 0 = {u 1 , u 2 , · · · , ut } and u 1 = u. Since u 1 is not adjacent to any u i for i = 2, · · · , t, it follows that | N (u 1 ) ∩ N (u i ) ∩ U | ≥ 1+(−1)k
1+(−1)k
k+ − | N (u 1 ) ∩ U | ≥ k + − 2k = 2k . There2 2 fore, there exists a vertex set U 11 = { v 1 , v 2 , · · · , v t −1 } such that v i ∈ N (u 1 ) ∩ N (u i +1 ) ∩ U for i = 1, 2, · · · , t − 1. Clearly,
U 11 ⊆ N (u ) ∩ U . Note that | N (u ) ∩ U | = d(u ) − | N (u ) ∩ U | ≥ δ(G ) − |[u , U ]| ≥ k + 2 − 2k = 2k + 2. This implies that there exists a vertex set U 1 ⊆ N (u ) ∩ U such that |U 1 | = 2k and U 11 ⊂ U 1 . It is easy to see that G [U 0 ∪ U 1 ] is a connected graph and |U 0 ∪ U 1 | < k. Since G [U ] is a connected graph with order |U | ≥ k + 1, there exists a vertex set W ⊂ U such that U 0 ∪ U 1 ⊂ W , | W | = k and G [ W ] is connected. Let v be an arbitrary vertex in U \ W . Then, by U 0 = U \ ( N (u ) ∩ U ) ⊂ W , we have that v ∈ N (u ) ∩ U . Since G is triangle-free, it follows that v is not adjacent to any vertex in N (u ) ∩ U , and hence v is not adjacent to any vertex in U 1 . This implies that |[ v , W ]| ≤ | W | − |U 1 | = k − 2k = 2k . Therefore, |[ W , U \ W ]| = v ∈U \W |[ v , W ]| ≤
2k |U \ W |. On the other hand, since U ∗ ⊆ U 0 , we have 1 that | N ( v ) ∩ U | ≥ k+ for any v ∈ U \ W . It follows that 2 |[U \ W , U ]| = v ∈U \W | N ( v ) ∩ U | ≥ k+2 1 |U \ W |. Therefore, |[ W , U \ W ]| < |[U \ W , U ]|, contradicting Lemma 2.1. The proof is completed. 2 By Theorem 2.8, we can easily show the complete bipartite graph K r ,s (r ≤ s) is at least super-λr −2 . Corollary 2.9. Let r, s be two integers and r ≤ s. Then for any integer k ≤ r − 2, the complete bipartite graph K r ,s is super-λk . In particular, K r ,s is super-λr −1 if r is an odd integer. Proof. It is easy to see that δ( K r ,s ) = r and | N (u ) ∩ N ( v )| = r for any pair of nonadjacent vertices u , v ∈ V ( K r ,s ). So, 1+(−1)k
for any integer k ≤ r − 2, | N (u ) ∩ N ( v )| = r ≥ k + 2 for any pair of nonadjacent vertices u , v ∈ V ( K r ,s ) and δ( K r ,s ) = r ≥ k + 1 − (−1)k . By Theorem 2.8, K r ,s is super-λk . In particular, if r is an odd integer and k = r − 1, 1+(−1)k
then | N (u ) ∩ N ( v )| = r = k + for any pair of non2 adjacent vertices u , v ∈ V ( K r ,s ) and δ( K r ,s ) = r > k + 1 − (−1)k . By Theorem 2.8, K r ,s is super-λr −1 . 2 In the following, we demonstrate an example to show Corollary 2.9 is the best possible for r is an even integer. Example 2.10. For any even integer r ≥ 2, the complete bipartite graph K r ,r has the minimum degree δ( K r ,r ) = r < (r − 1) + 1 − (−1)r −1 and fulfills | N (u ) ∩ N ( v )| = r ≥ r −1
for each pair of nonadjacent vertices (r − 1) + 1+(−21) u , v ∈ V ( K r ,r ). Let H = K r −2 , r be a subgraph of K r ,r . It 2
2
is easy to show ξr −1 ( K r ,r ) = |[ V ( H ), V ( K r ,r − H )]| =
r2 . 2
r2 . Let H = K r , r be an2 2 2 other subgraph of K r ,r and H = K r ,r − V ( H ). It can be 2 shown that |[ V ( H ), V ( H )]| = r2 and H = K r , r Thus 2 2 [ V ( H ), V ( H )] is a minimum k-restricted cut, but it is not
By Theorem 2.5, λr −1 ( K r ,r ) =
a set of edges incident to a certain connected subgraph of order r − 1, that is K r ,r is not super-λr −1 . On the other hand, by Corollary 2.9, K r ,r is super-λk for any k ≤ r − 2. A triangle-free graph G is maximal if the addition of any new edge creates a triangle. The maximal triangle-free graph is a type of interesting graph. Next, we show the triangle-free graphs satisfied the conditions of Theorem 2.8
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are maximal. A graph of diameter 2 is minimal if removing any edge increases the diameter. In [1], a sufficient and necessary condition is presented. Theorem 2.11. (See [1].) A triangle-free graph is maximal if and only if it is a minimum graph of diameter 2. Theorem 2.12. A triangle-free graph of diameter 2 is a minimum graph of diameter 2. Proof. Let G be a triangle-free graph of diameter 2, but not a minimum graph of diameter 2. Then there exists an edge uv ∈ E (G ) such that G − uv is also diameter 2. It follows that there is a (u , v )-path P of length 2 in G − uv. Then P uv is a triangle of G, a contradiction. 2 Combining Theorem 2.11 and Theorem 2.12, we can deduce a sufficient and necessary condition of maximal triangle-free graph immediately. Corollary 2.13. A triangle-free graph is maximal if and only if its diameter is 2. Proof. By Theorem 2.11, the necessity is obvious. Next, we shall show the sufficiency. Let G be a triangle-free graph of diameter 2. By Theorems 2.11 and 2.12, G is a maximal triangle-free graph. 2 By Corollary 2.13, we can deduce the triangle-free graphs satisfied the conditions of Theorem 2.8 are also maximal triangle-free graphs. 3. Conclusion In this paper, we consider the k-restricted connectivity of triangle-free graph and prove a triangle-free graph G to be super-λk if its any pair of nonadjacent vertices contain at least k +
1+(−1)k 2
common neighbors and the minimum
degree of G is more than k + 1 − (−1)k . As a corollary, we have obtained the complete bipartite graph K r ,s (r ≤ s) is
5 1+(−1)r
super-λk for any integer k ≤ r − 1 − , which is the 2 best possible when r is an even integer. Finally, we raise a few questions: What is the maximum k such that the complete graph K r ,s is super-λk ? Acknowledgements We would like to express our thanks to the referees for their helpful comments and suggestions. References [1] C. Barefoot, K. Casey, David Fisher, K. Fraughnaugh, F. Harary, Size in maximal triangle-free graphs and minimal graphs of diameter 2, Discrete Math. 138 (1995) 93–99. [2] J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, The Macmillan Press Ltd, New York, 1976. [3] P. Bonsma, N. Ueffing, L. Volkmann, Edge-cuts leaving components of order at least three, Discrete Math. 256 (2002) 431–439. [4] A.H. Esfahanian, S.L. Hakimi, On computing a conditional edgeconnectivity of a graph, Inf. Process. Lett. 27 (1988) 165–199. [5] J. Fàbrega, M.A. Fiol, Extraconnectivity of graphs with large girth, Discrete Math. 127 (1994) 163–170. [6] M.A. Fiol, On super-edge-connected digraphs and bipartite digraphs, J. Graph Theory 16 (1992) 545–555. [7] J.P. Ou, Edge cuts leaving components of order at least m, Discrete Math. 305 (2005) 365–371. [8] L. Shang, H.P. Zhang, Sufficient conditions for graphs to be λ -optimal and super-λ , Networks 49 (3) (2007) 234–242. [9] L. Shang, The high order restricted edge connectivity of graphs, Lanzhou University, Lanzhou, 2008. [10] S.Y. Wang, S.W. Lin, C.F. Li, Sufficient conditions for super k-restricted edge connectivity in graphs of diameter 2, Discrete Math. 309 (2009) 908–919. [11] S.Y. Wang, N. Zhao, Degree conditions for graphs to be maximally k-restricted edge connected and super k-restricted edge connected, Discrete Appl. Math. 184 (2015) 258–263. [12] S.Y. Wang, L. Zhang, S.W. Lin, k-Restricted edge connectivity in (p + 1)-clique-free graphs, Discrete Appl. Math. 181 (2015) 255–259. [13] J. Yuan, A.X. Liu, S.Y. Wang, Sufficient conditions for bipartite graphs to be super-λk -restricted edge-connected, Discrete Math. 309 (2009) 2886–2896. [14] J. Yuan, A.X. Liu, Sufficient conditions for λk -optimality in trianglefree graphs, Discrete Math. 310 (2010) 981–987. [15] Z. Zhang, J.J. Yuan, A proof of an inequality concerning k-restricted edge connectivity, Discrete Math. 304 (2005) 128–134. [16] Z. Zhang, J.J. Yuan, Degree conditions for restricted-edge-connectivity and isoperimetric-edge-connectivity to be optimal, Discrete Math. 307 (2007) 293–298.
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Information Processing Letters www.elsevier.com/locate/ipl
Sufficient conditions for triangle-free graphs to be super k-restricted edge-connected ✩ Jun Yuan a , Aixia Liu b,∗ a b
School of Applied Science, Taiyuan University of Science and Technology, Taiyuan, Shanxi 030024, People’s Republic of China School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 03006, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 11 June 2013 Received in revised form 7 April 2015 Accepted 3 September 2015 Available online xxxx Communicated by Tsan-sheng Hsu Keywords: Fault tolerance Restricted edge connectivity Super restricted edge connected connectivity Triangle-free graph
a b s t r a c t An edge cut S of a connected graph G = ( V , E ) is a k-restricted edge cut if every component of G − S contains at least k vertices. A graph is said to be super k-restricted edge-connected if every minimum k-restricted edge cut is a set of edges incident to a certain connected subgraph of order k. Let k be a positive integer, and let G be a connected triangle-free graph of order n ≥ 2k. In this paper, we prove that if the minimum degree
δ(G ) ≥ k + 1 − (−1)k and there are at least k +
common vertices in the neighbor sets of each pair of nonadjacent vertices in G, then G is super k-restricted edge-connected. © 2015 Elsevier B.V. All rights reserved.
1. Terminology and introduction For graph-theoretical terminology and notation not defined here, we follow [2]. In this paper, we consider only finite, undirected, and simple graphs G with the vertex set V (G ) and the edge set E (G ). For two vertices u , v ∈ V (G ), the distance between u and v is the length of the shortest path from u to v in G. The diameter of G is the maximum distance between two vertices of G. For two disjoint vertex set U 1 , U 2 ⊂ V (G ), the set of edges with one end in U 1 and the other in U 2 is denoted by [U 1 , U 2 ]. If U 1 = {u }, then [U 1 , U 2 ] is abbreviated as [u , U 2 ]. We use G [U ] to denote the subgraph of G induced by U ⊆ V (G ), and U = V (G ) \ U the complement of U . Let U be a vertex subset of V (G ). Define the open neighborhood N G (U ) of U in G as the set { v ∈ U : |[ v , U ]| ≥ 1}. In particular, if U = { v }, then N G (U ) is denoted by N G ( v ). The degree ✩ This work is supported by the National Science Foundation of China (61402317, 61303020) and the Natural Science Foundation of Shanxi Province (2012021001-2). Corresponding author. E-mail address: [email protected] (A. Liu).
*
http://dx.doi.org/10.1016/j.ipl.2015.09.005 0020-0190/© 2015 Elsevier B.V. All rights reserved.
1+(−1)k 2
d G ( v ) of a vertex v is the number of vertices in N G ( v ). Let δ(G ) denote the minimum vertex degree in G. For N G (U ) and d G ( v ), we usually omit the subscript for the graph when no confusion arises. An edge subset M of E (G ) is called a matching in G if no two edges of M are adjacent in G. M is a maximum matching if G has no matching M with | M | > | M |. A subset S of V (G ) is called an independent set of G if no two vertices of S are adjacent in G. An independent set S is maximum if G has no independent set S with | S | > | S |. The number of vertices in a maximum independent set of G is called the independence number of G and is denoted by α (G ). A subset K of V (G ) such that every edge of G has at least one end in K is called a covering of G. A covering K is minimum if G has no covering K with | K | < | K |. The number of vertices in a minimum covering of G is called the covering number of G and is denoted by β(G ). A network can be conveniently modeled as a graph. One fundamental consideration in the design of networks is reliability. A classic measure of the reliability of a network is the edge connectivity λ(G ). In general, the larger λ(G ) is, the more reliable the network is. A more refined measure known as restricted edge connectivity was pro-
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posed by Esfahanian and Hakimi [4], which was further generalized to k-restricted edge connectivity by Fàbrega and Fiol [5] (called k-extra edge connectivity in their paper). For a connected graph G, an edge set S ⊂ E (G ) is said to be a k-restricted edge cut if G − S is disconnected and every component of G − S contains at least k vertices. The cardinality of the minimum k-restricted edge cut of G, denoted as λk (G ), is called the k-restricted edge connectivity of G. It should be emphasized that not all connected graphs have k-restricted edge cuts. A connected graph G is said to be λk -connected if G has a k-restricted edge cut. Sufficient conditions for graphs to be λk -connected were shown by several authors [3,4,7,15]. According to recent studies on the k-restricted edge connectivity of G, it seems that the larger λk (G ) is, the more reliable the network is. So we expect λk (G ) to be as large as possible. Define ξk (G ) = min{|[U , U ]| : U ⊂ V (G ), |U | = k ≥ 1 and G [U ] is connected}. It has been shown that λk ≤ ξk holds for many graphs [3,7,15]. A connected graph G with λk (G ) = ξk (G ) is said to be λk -optimal. Furthermore, if every minimum k-restricted edge cut is a set of edges incident to a certain connected subgraph of order k, then G is said to be super k-restricted edge-connected, in short, super-λk . In particular, super-λ1 and super-λ2 are denoted as super-λ and super-λ , respectively. By definition, if G is super-λk , then G is λk -optimal. However, the converse is not true. For example, a cycle of length n ≥ 2k + 2 is λk -optimal, but not super-λk . Investigations on the restricted edge connectivity of graphs were made by several authors, for example, by [8,9,11–14,16]. In 2009, Wang and Lin [10] gave a neighborhood condition for graphs with diameter 2 to be super-λk graphs. Theorem 1.1. (See [10].) Let k be a positive integer and let G be a graph with order at least 2k. If | N (u ) ∩ N ( v )| ≥ 2k + 1 for all pairs u, v of nonadjacent vertices, then G is super-λk . For k = 2, 3, Shang and Zhang [8,9] once showed the following neighborhood conditions for graphs to be super-λk . Theorem 1.2. (See [8].) Let G be a connected triangle-free graph with order ν ≥ 4 and different from K 2,ν −2 . If | N (u ) ∩ N ( v )| ≥ 2 for all pairs u, v of nonadjacent vertices, then G is super-λ2 . Theorem 1.3. (See [9].) Let G be a triangle-free graph with order
ν ≥ 6 and minimum degree δ(G ) ≥ 5. If | N (u ) ∩ N ( v )| ≥ 2 for all pairs u , v of nonadjacent vertices, then G is super-λ3 . By contrast Theorems 1.2, 1.3 with Theorem 1.1, it is easy to see that the requirement of Theorem 1.1 is too high to determine a triangle-free graph to be super-λk for a genereal integer k. However, for a general integer k, the neighborhood condition for triangle-free graphs to be super-λk have not been given out yet. In this paper, we show a neighborhood condition for triangle-free graphs to be super-λk for general integer k. 2. Main result A vertex set U ⊂ V (G ) is called a λk -fragment if [U , U ] is a k-restricted edge cut of G with |[U , U ]| = λk (G ). For
a λk -fragment U , it is easy to see that G [U ] and G [U ] are both connected. In [13], the authors have proved that: Lemma 2.1. (See [13].) Let G be a λk -connected graph with λk (G ) ≤ ξk (G ) and let U be a λk -fragment of G. There exists no connected subgraph H of order k in G [U ] such that
|[ V ( H ), U \ V ( H )]| < |[U \ V ( H ), U ]|. Lemma 2.2. Let G be a λk -optimal triangle-free graph with minimum degree δ(G ) ≥ k + 1 − (−1)k . If G is not super-λk , then for any λk -fragment U of G with |U | ≥ k + 1 and |U | ≥ k + 1, the set
k U ∗ = { v ∈ U : |[ v , U ]| ≤ } = ∅. 2 Proof. Note that |[ v , U ]| is integer. It is sufficient to proof that U ∗ = { v ∈ U : |[ v , U ]| ≤ 2k } = ∅. Suppose on the contrary that U ∗ = { v ∈ U : |[ v , U ]| ≤ 2k } = ∅, that is
|[ v , U ]| ≥
k+1
(2.1)
2
for any v ∈ U . Let W be a vertex set of k vertices in U satisfying the following conditions: (i) G [ W ] is connected; (ii) f ( W ) = max{|[ v , W ]| : v ∈ U \ W } is as small as possible; denote the number by f 0 ; (iii) g ( W ) = |{ v : v ∈ U \ W , |[ v , W ]| = f 0 }| is as small as possible; denote the number by g 0 . By (2.1),
|[U \ W , U ]| =
|[ v , U ]| ≥
k+1
v ∈U \ W
2
|U \ W |.
(2.2)
Consider the following two cases. Case 1. f 0 ≤ 2k .
In this case, for any v ∈ U \ W , we have that |[ v , W ]| ≤ 2k .
This implies that |[ W ,U \ W ]| =
v ∈U \ W
|[ v , W ]| ≤ 2k |U \ W |.
Combining this with (2.2), we have that |[ W , U \ W ]| <
|[U \ W , U ]|, contradicting to Lemma 2.1. Case 2. f 0 > 2k .
1 We first show f 0 ≥ 2k + 1. Suppose that f 0 = k+ . By 2 the definition of f 0 , we have that
|[ W , U \ W ]| ≤ f 0 |U \ W | =
k+1 2
|U \ W |.
(2.3)
Combining this with (2.2), we have |[ W , U \ W ]| ≤ |[U \ W , U ]|. By Lemma 2.1, |[ W , U \ W ]| = |[U \ W , U ]|. This implies that the inequalities of (2.2) and (2.3) must be 1 equalities, that is |[ W , U \ W ]| = k+ |U \ W | = |[U \ W , U ]|. 2 1 It follows that |[ v , W ]| = |[ v , U ]| = k+ for any v ∈ U \ W . 2
Combining this with δ(G ) ≥ k + 1 − (−1)k , there exist at least one edge in G [U \ W ], say xy. Since G is triangle-free, it follows that N (x) ∩ N ( y ) = ∅. So d G [ W ] (x) + d G [ W ] ( y ) ≤ k. 1 On the other hand, by |[ v , W ]| = k+ for any v ∈ U \ W , 2
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we can conclude that d G [ W ] (x) + d G [ W ] ( y ) = k + 1, a contradiction. Therefore,
f0 ≥
k 2
+ 1.
(2.4)
Next, we shall construct a new connected subgraph W of order k such that f ( W ) < f ( W ) or f ( W ) = f ( W ) and g ( W ) < g ( w ). Let u be a vertex in U \ W such that |[u , W ]| = f 0 , that is | N (u ) ∩ W | = f 0 Since G is trianglefree, N (u ) ∩ W is an independent set of G. Let T be a spanning tree of G [ W ], and let S = { w : w ∈ N (u ) ∩ W , d T ( w ) = 1}. Clearly, the edge number ε ( T ) = k − 1 ≥ 2(| N (u ) ∩ W | − | S |) + | S | = 2 f 0 − | S | > k − | S |, and hence | S | > 1. Let w be a vertex of S. Then construct a new subgraph W such that W = W − { w } + {u }. Therefore, G [ W ] is connected and | W | = k. Since G is trianglefree, N ( w ) ∩ N (u ) = ∅. It follows that | N ( w ) ∩ W | ≤ | W | − | N (u ) ∩ W | = k − f 0 < k − 2k = 2k . Note that | N ( w ) ∩ W | is an integer. By the choice of w, we have that
k
| N ( w ) ∩ W | = | N ( w ) ∩ W | + 1 ≤ . 2
(2.5)
| N (u ) ∩ N ( v )| ≥ k for each pair of nonadjacent vertices u , v ∈ V (G ), then G is λk -optimal. Theorem 2.6. (See [1, Theorem 5.3, p. 74].) In a bipartite graph, the number of edges in a maximum matching is equal to the covering number. Theorem 2.7. (See [1, Corollary 7.1, p. 101].) Let G be a graph of order n, and let α (G ) and β(G ) be the independence number and the covering number. Then α (G ) + β(G ) = n. Theorem 2.8. Let k be a positive integer, and let G be a trianglefree graph of order n ≥ 2k and minimum degree δ(G ) ≥ k + 1 − (−1)k . If
| N (u ) ∩ N ( v )| ≥ k +
1 + (−1)k 2
(2.6)
Let v be a vertex in U \ ( W ∪ {u }). If uv ∈ E (G ), then, since G is triangle-free, we have that N (u ) ∩ N ( v ) = ∅. This implies that v is not adjacent to w and | N ( v ) ∩ W | ≤ | W | − | N (u ) ∩ W | = k − f 0 < 2k . Combining this with (2.4), we have that
|[ v , W ]| = |[ v , W ]| + 1 = |N (v ) ∩ W | + 1 <
k 2
+ 1 ≤ f 0.
(2.7)
If uv ∈ / E (G ), then
|[ v , W ]| ≤ |[ v , W ]|.
(2.8)
Combining (2.6), (2.7), and (2.8) with the definition of f 0 , we can conclude that 1, then f ( W )
(a) if g 0 = < f 0 , which contradicts condition (ii); (b) if g 0 ≥ 2, then f ( W ) = f 0 and g ( W ) = g 0 − 1, which contradicts condition (iii). The proof is completed.
Theorem 2.5. (See [14].) Let k be a positive integer, and let G be a triangle-free graph of order n ≥ 2k. If
for each pair of nonadjacent vertices u , v ∈ V (G ), then G is super-λk .
Combining (2.4) and (2.5), we have that
|N ( w ) ∩ W | < f 0 .
3
2
The following results play important roles in the proof of Theorem 2.8.
1+(−1)k
Proof. By the assumption that | N (u ) ∩ N ( v )| ≥ k + 2 for each pair of nonadjacent vertices u , v ∈ V (G ), the diameter of G is 2. If k = 1, then by δ(G ) ≥ k + 1 − (−1)k = 3 and Theorem 2.3, G is super-λ1 . If k = 2, then by assump1+(−1)k
= 3 for each tion, we have | N (u ) ∩ N ( v )| ≥ k + 2 pair of nonadjacent vertices u , v ∈ V (G ). It follows that G = K 2,n−2 . By Theorem 2.4, G is super-λ2 . Suppose k ≥ 3. By Theorem 2.5, G is λk -optimal and hence λk (G ) = ξk (G ). Suppose to the contrary that G is not super-λk . Then there exists a λk -fragment U such that |U | ≥ k + 1 and |U | ≥ k + 1. By Lemma 2.2, the set U ∗ = { v ∈ U : |[ v , U ]| ≤ 2k } = ∅, U ∗ = { v ∈ U : |[ v , U ]| ≤ 2k } = ∅. Let u be an arbitrary
element in U ∗ and v an arbitrary element in U ∗ . Claim 1. u is adjacent to v. Otherwise, by assumption, we have that
k+
1 + (−1)k 2
≤ | N (u ) ∩ N ( v )| = | N (u ) ∩ N ( v ) ∩ U | + | N (u ) ∩ N ( v ) ∩ U | ≤ | N ( v ) ∩ U | + | N (u ) ∩ U | k
≤ 2 , 2
Theorem 2.3. (See [6].) Let G be a graph of order n and minimum degree δ . Then G is super-λ if the diameter of G is 2 and G contains no the complete graph K δ with all its vertices of degree δ . Theorem 2.4. (See [8].) Let G be a connected triangle-free graph of order n ≥ 4 and different from K 2,n−2. If
| N (u ) ∩ N ( v )| ≥ 2 for all pairs u, v of nonadjacent vertices, then G is super-λ .
a contradiction. The proof of Claim 1 is completed. Claim 2. |U ∗ | ≤ 2k . By Claim 1, we have that U ∗ ⊆ N ( v ). It follows that U ∗ ⊆ N ( v ) ∩ U , and hence |U ∗ | ≤ | N ( v ) ∩ U | ≤ 2k . The proof of Claim 2 is completed. Combining Claim 1 with the fact that G is triangle-free, it can be deduced that U ∗ is an independent set of G. We consider the following two cases.
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4
Case 1. There is a vertex u 1 ∈ U ∗ such that |U \ ( N (u 1 ) ∩ U )| ≥ 2k . Since U ∗ is an independent set of G, we can deduce that U ∗ ⊆ U \ ( N (u 1 ) ∩ U ). By |U \ ( N (u 1 ) ∩ U )| ≥ 2k and Claim 2, there exists a set U 0 ⊆ U \ ( N (u 1 ) ∩ U ) such that U ∗ ⊆ U 0 and |U 0 | = 2k . Let U 0 = {u 1 , u 2 , · · · , u k }. 2
Clearly, u 1 is not adjacent to any vertex in U 0 \ {u 1 }. By assumption, we have that | N (u 1 ) ∩ N (u i ) ∩ U | = | N (u 1 ) ∩ 1) N (u i )| − | N (u 1 ) ∩ N (u i ) ∩ U | ≥ k + 1+(− − | N (u 1 ) ∩ U | ≥ 2 k
for i = 2, · · · , 2k . {v 1, v 2, · · · , v k } 2
k 2
Therefore, there exists a vertex set U 1 = such that v i ∈ N (u 1 ) ∩ N (u i +1 ) ∩ U for
i = 1, · · · , 2k − 1 and v k , v k ∈ N (u 1 ) ∩ N (u k ) ∩ U . 2 2 2 Let W = U 0 ∪ U 1 . Clearly, G [ W ] is connected. We shall show that the independent number α (G [ W ]) ≤ 2k . Let H = G [ W ] − E (G [U 0 ]). Then H is a bipartite graph with bipartition (U 0 , U 1 ). Clearly, the edge set {u i v i −1 : i = 2, · · · , 2k } ∪ {u 1 v k } is a maximum match2
ing of H (define {u i v i −1 : i = 2, · · · , 2k } = ∅ when k = 3). By Theorem 2.6 and Theorem 2.7, we have that α ( H ) = | V ( H )| − 2k = 2k . It follows that
k
α (G [ W ]) ≤ α ( H ) = .
(2.9)
2
Let v be an arbitrary vertex of U \ W . Since G is triangle-free, we have that N ( v ) ∩ W is an independent set of G. By (2.9), we have that | N ( v ) ∩ W | ≤ 2k . It follows
| N ( v ) ∩ W | ≤ 2k |U \ W |. On the other hand, since U ∗ ⊆ U 0 , we have that | N ( v ) ∩ U | ≥ k+2 1 for any v ∈ U \ W . It follows that |[U \ W , U ]| = k+1 v ∈U \ W | N ( v ) ∩ U | ≥ 2 |U \ W |. So, that |[ W , U \ W ]| =
v ∈U \ W
k
|[ W , U \ W ]| ≤ |U \ W | ≤ 2 ≤ |[U \ W , U ]|.
k+1 2
|U \ W |
If k is even, then |[ W , U \ W ]| < |[U \ W , U ]|. On the other hand, since λk (G ) = ξk (G ), by Lemma 2.1, we have that |[ W , U \ W ]| ≥ |[U \ W , U ]|, a contradiction. So, suppose k is odd. Then we have |[ W , U \ W ]| ≤ |[U \ W , U ]|. Since λk (G ) = ξk (G ), by Lemma 2.1, we have that |[ W , U \ W ]| ≥ |[U \ W , U ]| and hence |[ W , U \ W ]| = |[U \ W , U ]|. It im1 plies that | N ( v ) ∩ W | = | N ( v ) ∩ U | = k+ for any v ∈ U \ W . 2 So, we deduce that | N ( v ) ∩ N ( w ) ∩ W | ≥ | N ( v ) ∩ W | + | N ( w ) ∩ W | − | W | = 1 for any v , w ∈ U \ W . Note G is triangle-free. We have U \ W is an independent set and hence for any v ∈ U \ W , d( v ) = | N ( v ) ∩ W | + | N ( v ) ∩ U | = k + 1. It implies that δ(G ) ≤ k + 1. On the other hand, since k is odd, by assumption, δ(G ) ≥ k + 1 − (−1)k = k + 2, a contradiction. Case 2. |U \ ( N (u ) ∩ U )| < 2k for any u ∈ U ∗ . Let u be an arbitrary vertex of U ∗ , and let U 0 = U \ ( N (u ) ∩ U ). Note that U ∗ is an independent set of G and u ∈ U ∗ . By the definition of U 0 , U ∗ ⊆ U 0 . Let U 0 = {u 1 , u 2 , · · · , ut } and u 1 = u. Since u 1 is not adjacent to any u i for i = 2, · · · , t, it follows that | N (u 1 ) ∩ N (u i ) ∩ U | ≥ 1+(−1)k
1+(−1)k
k+ − | N (u 1 ) ∩ U | ≥ k + − 2k = 2k . There2 2 fore, there exists a vertex set U 11 = { v 1 , v 2 , · · · , v t −1 } such that v i ∈ N (u 1 ) ∩ N (u i +1 ) ∩ U for i = 1, 2, · · · , t − 1. Clearly,
U 11 ⊆ N (u ) ∩ U . Note that | N (u ) ∩ U | = d(u ) − | N (u ) ∩ U | ≥ δ(G ) − |[u , U ]| ≥ k + 2 − 2k = 2k + 2. This implies that there exists a vertex set U 1 ⊆ N (u ) ∩ U such that |U 1 | = 2k and U 11 ⊂ U 1 . It is easy to see that G [U 0 ∪ U 1 ] is a connected graph and |U 0 ∪ U 1 | < k. Since G [U ] is a connected graph with order |U | ≥ k + 1, there exists a vertex set W ⊂ U such that U 0 ∪ U 1 ⊂ W , | W | = k and G [ W ] is connected. Let v be an arbitrary vertex in U \ W . Then, by U 0 = U \ ( N (u ) ∩ U ) ⊂ W , we have that v ∈ N (u ) ∩ U . Since G is triangle-free, it follows that v is not adjacent to any vertex in N (u ) ∩ U , and hence v is not adjacent to any vertex in U 1 . This implies that |[ v , W ]| ≤ | W | − |U 1 | = k − 2k = 2k . Therefore, |[ W , U \ W ]| = v ∈U \W |[ v , W ]| ≤
2k |U \ W |. On the other hand, since U ∗ ⊆ U 0 , we have 1 that | N ( v ) ∩ U | ≥ k+ for any v ∈ U \ W . It follows that 2 |[U \ W , U ]| = v ∈U \W | N ( v ) ∩ U | ≥ k+2 1 |U \ W |. Therefore, |[ W , U \ W ]| < |[U \ W , U ]|, contradicting Lemma 2.1. The proof is completed. 2 By Theorem 2.8, we can easily show the complete bipartite graph K r ,s (r ≤ s) is at least super-λr −2 . Corollary 2.9. Let r, s be two integers and r ≤ s. Then for any integer k ≤ r − 2, the complete bipartite graph K r ,s is super-λk . In particular, K r ,s is super-λr −1 if r is an odd integer. Proof. It is easy to see that δ( K r ,s ) = r and | N (u ) ∩ N ( v )| = r for any pair of nonadjacent vertices u , v ∈ V ( K r ,s ). So, 1+(−1)k
for any integer k ≤ r − 2, | N (u ) ∩ N ( v )| = r ≥ k + 2 for any pair of nonadjacent vertices u , v ∈ V ( K r ,s ) and δ( K r ,s ) = r ≥ k + 1 − (−1)k . By Theorem 2.8, K r ,s is super-λk . In particular, if r is an odd integer and k = r − 1, 1+(−1)k
then | N (u ) ∩ N ( v )| = r = k + for any pair of non2 adjacent vertices u , v ∈ V ( K r ,s ) and δ( K r ,s ) = r > k + 1 − (−1)k . By Theorem 2.8, K r ,s is super-λr −1 . 2 In the following, we demonstrate an example to show Corollary 2.9 is the best possible for r is an even integer. Example 2.10. For any even integer r ≥ 2, the complete bipartite graph K r ,r has the minimum degree δ( K r ,r ) = r < (r − 1) + 1 − (−1)r −1 and fulfills | N (u ) ∩ N ( v )| = r ≥ r −1
for each pair of nonadjacent vertices (r − 1) + 1+(−21) u , v ∈ V ( K r ,r ). Let H = K r −2 , r be a subgraph of K r ,r . It 2
2
is easy to show ξr −1 ( K r ,r ) = |[ V ( H ), V ( K r ,r − H )]| =
r2 . 2
r2 . Let H = K r , r be an2 2 2 other subgraph of K r ,r and H = K r ,r − V ( H ). It can be 2 shown that |[ V ( H ), V ( H )]| = r2 and H = K r , r Thus 2 2 [ V ( H ), V ( H )] is a minimum k-restricted cut, but it is not
By Theorem 2.5, λr −1 ( K r ,r ) =
a set of edges incident to a certain connected subgraph of order r − 1, that is K r ,r is not super-λr −1 . On the other hand, by Corollary 2.9, K r ,r is super-λk for any k ≤ r − 2. A triangle-free graph G is maximal if the addition of any new edge creates a triangle. The maximal triangle-free graph is a type of interesting graph. Next, we show the triangle-free graphs satisfied the conditions of Theorem 2.8
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are maximal. A graph of diameter 2 is minimal if removing any edge increases the diameter. In [1], a sufficient and necessary condition is presented. Theorem 2.11. (See [1].) A triangle-free graph is maximal if and only if it is a minimum graph of diameter 2. Theorem 2.12. A triangle-free graph of diameter 2 is a minimum graph of diameter 2. Proof. Let G be a triangle-free graph of diameter 2, but not a minimum graph of diameter 2. Then there exists an edge uv ∈ E (G ) such that G − uv is also diameter 2. It follows that there is a (u , v )-path P of length 2 in G − uv. Then P uv is a triangle of G, a contradiction. 2 Combining Theorem 2.11 and Theorem 2.12, we can deduce a sufficient and necessary condition of maximal triangle-free graph immediately. Corollary 2.13. A triangle-free graph is maximal if and only if its diameter is 2. Proof. By Theorem 2.11, the necessity is obvious. Next, we shall show the sufficiency. Let G be a triangle-free graph of diameter 2. By Theorems 2.11 and 2.12, G is a maximal triangle-free graph. 2 By Corollary 2.13, we can deduce the triangle-free graphs satisfied the conditions of Theorem 2.8 are also maximal triangle-free graphs. 3. Conclusion In this paper, we consider the k-restricted connectivity of triangle-free graph and prove a triangle-free graph G to be super-λk if its any pair of nonadjacent vertices contain at least k +
1+(−1)k 2
common neighbors and the minimum
degree of G is more than k + 1 − (−1)k . As a corollary, we have obtained the complete bipartite graph K r ,s (r ≤ s) is
5 1+(−1)r
super-λk for any integer k ≤ r − 1 − , which is the 2 best possible when r is an even integer. Finally, we raise a few questions: What is the maximum k such that the complete graph K r ,s is super-λk ? Acknowledgements We would like to express our thanks to the referees for their helpful comments and suggestions. References [1] C. Barefoot, K. Casey, David Fisher, K. Fraughnaugh, F. Harary, Size in maximal triangle-free graphs and minimal graphs of diameter 2, Discrete Math. 138 (1995) 93–99. [2] J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, The Macmillan Press Ltd, New York, 1976. [3] P. Bonsma, N. Ueffing, L. Volkmann, Edge-cuts leaving components of order at least three, Discrete Math. 256 (2002) 431–439. [4] A.H. Esfahanian, S.L. Hakimi, On computing a conditional edgeconnectivity of a graph, Inf. Process. Lett. 27 (1988) 165–199. [5] J. Fàbrega, M.A. Fiol, Extraconnectivity of graphs with large girth, Discrete Math. 127 (1994) 163–170. [6] M.A. Fiol, On super-edge-connected digraphs and bipartite digraphs, J. Graph Theory 16 (1992) 545–555. [7] J.P. Ou, Edge cuts leaving components of order at least m, Discrete Math. 305 (2005) 365–371. [8] L. Shang, H.P. Zhang, Sufficient conditions for graphs to be λ -optimal and super-λ , Networks 49 (3) (2007) 234–242. [9] L. Shang, The high order restricted edge connectivity of graphs, Lanzhou University, Lanzhou, 2008. [10] S.Y. Wang, S.W. Lin, C.F. Li, Sufficient conditions for super k-restricted edge connectivity in graphs of diameter 2, Discrete Math. 309 (2009) 908–919. [11] S.Y. Wang, N. Zhao, Degree conditions for graphs to be maximally k-restricted edge connected and super k-restricted edge connected, Discrete Appl. Math. 184 (2015) 258–263. [12] S.Y. Wang, L. Zhang, S.W. Lin, k-Restricted edge connectivity in (p + 1)-clique-free graphs, Discrete Appl. Math. 181 (2015) 255–259. [13] J. Yuan, A.X. Liu, S.Y. Wang, Sufficient conditions for bipartite graphs to be super-λk -restricted edge-connected, Discrete Math. 309 (2009) 2886–2896. [14] J. Yuan, A.X. Liu, Sufficient conditions for λk -optimality in trianglefree graphs, Discrete Math. 310 (2010) 981–987. [15] Z. Zhang, J.J. Yuan, A proof of an inequality concerning k-restricted edge connectivity, Discrete Math. 304 (2005) 128–134. [16] Z. Zhang, J.J. Yuan, Degree conditions for restricted-edge-connectivity and isoperimetric-edge-connectivity to be optimal, Discrete Math. 307 (2007) 293–298.