k -restricted edge connectivity in (p+1) -clique-free graphs

Discrete Applied Mathematics 181 (2015) 255–259

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k-restricted edge connectivity in (p + 1)-clique-free graphs✩ Shiying Wang a,b,∗ , Lei Zhang a , Shangwei Lin a a

School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, PR China

b

College of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan 453007, PR China

article

info

Article history: Received 16 December 2013 Received in revised form 3 July 2014 Accepted 9 October 2014 Available online 1 November 2014 Keywords: Interconnection network Graph Restricted edge connectivity Clique

abstract Let G be a graph with vertex set V (G) and edge set E (G). An edge subset S ⊆ E (G) is called a k-restricted edge cut if G − S is not connected and every component of G − S has at least k vertices. The k-restricted edge connectivity of a connected graph G, denoted by λk (G), is defined as the cardinality of a minimum k-restricted edge cut. Let [X , X¯ ] denote the set of edges between a vertex set X ⊂ V (G) and its complement X¯ = V (G)\X . A vertex set X ⊂ V (G) is called a λk -fragment if [X , X¯ ] is a minimum k-restricted edge cut of G. Let ξk (G) = min{|[X , X¯ ]| : |X | = k, G[X ] is connected}. In this work, we give a lower bound on the cardinality of λk -fragments of a graph G satisfying λk (G) < ξk (G) and containing no (p + 1)-cliques. As a consequence of this result, we show a sufficient condition for a graph G with λk (G) = ξk (G). © 2014 Elsevier B.V. All rights reserved.

1. Terminology and introduction The topology of an interconnection network can be conveniently modeled as a graph in which the vertices represent the processors and the edges represent communication links in the network. The edge connectiv ity of a graph G, denoted by λ(G), is defined to be the minimum number of edges among all edge cuts of G, which is a classical measurement of the fault tolerance of a network. In general, the larger the λ(G) is, the more reliable the network is. As a more refined index than the edge connectivity, the restricted edge connectivity was proposed by Esfahanian and Hakimi in [5]. There is a significant amount of research on restricted edge connectivity [1,8,10,15,17]. In [6], Fàbrega and Fiol proposed the concept of k-restricted edge connectivity, which generalized the edge connectivity and the restricted edge connectivity. Let G be a connected graph with vertex set V (G) and edge set E (G) and let k be a positive integer. An edge set S ⊆ E (G) is a k-restricted edge cut if G − S is disconnected and every component of G − S has at least k vertices. The minimum number of edges in a k-restricted edge cut of G is called the k-restricted edge connectivity of G and denoted λk (G). A k-restricted edge cut S is a λk -cut if |S | = λk (G). It should be pointed out that not all connected graphs have a k-restricted edge cut. A connected graph G is said to be λk -connected if G has a k-restricted edge cut. As an important index to estimate the reliability of networks, the k-restricted edge connectivity recently has received a good deal of attention [2,7,11,13,18,19]. Let [X , X¯ ] denote the set of edges between a vertex set X ⊂ V (G) and its complement X¯ = V (G)\X . The following observation has been shown in [18]: if S is a λk -cut of a connected graph G, then there exists X ⊂ V (G) such that n − k ≥ |X | ≥ k, the induced subgraphs G[X ] and G[X¯ ] are both connected, and S = [X , X¯ ]. The optimization of λk (G) requires an upper bound. Let ξk (G) = min{|[X , X¯ ]| : |X | = k, G[X ] is connected}. It has been shown that λk (G) ≤ ξk (G)

✩ This work is supported by the National Natural Science Foundation of China (61370001, 61202017) and the Doctoral Fund of Ministry of Education of China (20111401110005). ∗ Corresponding author at: School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, PR China. E-mail addresses: [email protected] (S. Wang), [email protected] (L. Zhang), [email protected] (S. Lin).

http://dx.doi.org/10.1016/j.dam.2014.10.008 0166-218X/© 2014 Elsevier B.V. All rights reserved.

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S. Wang et al. / Discrete Applied Mathematics 181 (2015) 255–259

holds for many graphs G (see [2,4,7,12,14,19]). A connected graph G is called a λk -optimal graph if λk (G) = ξk (G). Sufficient conditions for λk -optimal graphs were given by several authors [2,9,18]. For graph-theoretical terminology and notation not defined here we follow [3]. We consider finite, undirected and simple graphs. The number of vertices of a graph is its order. The set of vertices adjacent to a vertex v in a graph G is denoted by NG (v). The degree of a vertex v in a graph G, denoted by dG (v), is the number of edges of G incident with v . Let δ(G) = min{dG (u) : u ∈ V (G)}. A clique of a graph is a set of mutually adjacent vertices. A graph G is called a (p + 1)-clique-free graph if it contains no (p + 1)-cliques. Let G be a λk -connected graph, and let [X , X¯ ] be a λk -cut. Then X is called a λk -fragment of G. Define rk (G) = min{|X | : X is a λk -fragment of G}. Obviously, k ≤ rk (G) ≤ ⌊ 2n ⌋. A λk -fragment X is called a λk -atom of G if |X | = rk (G). In 2007, Zhang and Yuan [20] gave a lower bound on the cardinality of λk -atoms of a graph G with λk (G) < ξk (G). Theorem 1.1 ([20]). Let G be a λk -connected graph. If λk (G) < ξk (G), then rk (G) ≥ max{k + 1, δ(G) − k + 2}. In 2012, Holtkamp et al. [9] studied a similar problem for (p + 1)-clique-free graphs, and presented the following. Theorem 1.2 ([9]). Let G be a λk -connected and 3-clique-free graph. If λk (G) < ξk (G), then rk (G) ≥ max{k + 1, 2δ(G) − k + 1}. Conjecture 1.3 ([9]). Let G be a λk -connected and (p + 1)-clique-free graph. If λk (G) < ξk (G), then rk (G) ≥ max{k + 1, p δ(G) − k + p−1 1 }. p−1 In this paper, we prove that rk (G) ≥ max{k + 1, p−1 δ(G) − k − 1} for any (p + 1)-clique-free graph G with λk (G) < ξk (G). p

2. Main results A k-partite graph is one whose vertex set can be partitioned into k subsets, or partite sets, in such a way that no edge has both ends in the same partite set. A k-partite graph with partite sets V1 , V2 , . . . , Vk−1 and Vk is balanced if ||Vi | − |Vj || ≤ 1 for any 1 ≤ i, j ≤ k. A balanced k-partite graph is complete if any two vertices in different partite sets are adjacent. A balanced complete k-partite graph on n vertices is called a Turán graph and denoted Tk,n . Theorem 2.1 ([16]). Let G be a (p + 1)-clique-free graph of order n. Then |E (G)| ≤ |E (Tp,n )|, with equality if and only if G is isomorphic to Tp,n . Set n = ps + r, where 0 ≤ r < p. It is not difficult to verify that

|E (Tp,n )| =

n2 − sn − r − rs 2

≤

(p − 1)n2 2p

.

(1)

Let G be a (p + 1)-clique-free graph of order n. Let x0 , x1 , . . . , xn−1 be n vertices, and let Vj = {xi : 0 ≤ i ≤ n − 1 and i ≡ j(mod p)}, where j = 0, . . . , p − 1. Let T be a complete p-partite graph with partite sets V0 , V1 , . . . , Vp−2 and Vp−1 . Clearly, T is isomorphic to Tp,n . Set X = {x0 , . . . , xk−1 } and Y = {xk , . . . , xn−1 }. It is easy to see that the induced subgraphs T [X ] and T [Y ] are isomorphic to Tp,k and Tp,n−k , respectively. Let α = |E (Tp,n )| − |E (G)|. By Theorem 2.1, α ≥ 0. Let G∗ be a graph obtained from T by deleting α edges such that |E (G∗ [X ])| is minimum. If α ≥ |E (Tp,k )| = |E (T [X ])|, then, by the minimality of |E (G∗ [X ])|, all edges in T [X ] are deleted and so E (G∗ [X ]) = ∅. If α < |E (Tp,k )| = |E (T [X ])|, then, by the minimality of |E (G∗ [X ])|, the edges deleted are all in T [X ], which implies that

α = |E (Tp,k )| − |E (G∗ [X ])| = |E (T [X ])| − |E (G∗ [X ])|. Clearly, |E (G∗ )| = |E (G)|. Lemma 2.2. Let G be a (p + 1)-clique-free graph of order n which contains at least a connected subgraph of order k, and let H be a connected subgraph of order k which contains as many edges as possible. (a) If k ≤ p, then

|E (G)| ≤ |E (Tp,n )| + |E (H )| −

(k − 1)k 2

.

(b) If k > p ≥ 3, then

|E (G)| ≤ |E (Tp,n )| + |E (H )| − |E (Tp,k )|. Proof. Let H ∗ = G∗ [X ]. Notice that |E (Tp,n )| − |E (G∗ )| ≥ |E (Tp,k )| − |E (H ∗ )|. Therefore, if k ≤ p, then

|E (G∗ )| ≤ |E (Tp,n )| + |E (H ∗ )| −

(k − 1)k 2

;

(2)

and if k > p ≥ 3, then

|E (G∗ )| ≤ |E (Tp,n )| + |E (H ∗ )| − |E (Tp,k )|.

(3)

S. Wang et al. / Discrete Applied Mathematics 181 (2015) 255–259

257

(k−1)k

Suppose that |E (H ∗ )| ≤ |E (H )|. If k ≤ p, then, by (2), we have |E (G)| = |E (G∗ )| ≤ |E (Tp,n )| + |E (H )| − 2 . If k > p ≥ 3, then, by (3), we have |E (G)| = |E (G∗ )| ≤ |E (Tp,n )| + |E (H )| − |E (Tp,k )|. Therefore, it remains to consider the case |E (H ∗ )| > |E (H )|. In this case, we have E (H ∗ ) ̸= ∅. By the minimality of |E (H ∗ )|, we have that α < |E (Tp,k )| and the edges deleted are all in T [X ]. It follows that G∗ − X is isomorphic to Tp,n−k and for any x ∈ Y ,

|NG∗ (x) ∩ X | ≥ k −

  k

p

.

(4)

By Theorem 2.1, |E (G − V (H ))| ≤ |E (Tp,n−k )| = |E (G∗ − X )|. Since |E (H )| + |E (G − V (H ))| + |[V (H ), V (H )]| =  |E (G)| = |E (G∗ )| = |E (H ∗ )| + |E (G∗ − X )| + |[X , Y ]|, we have |[X , Y ]| < |[V (H ), V (H )]|, that is, x∈Y |NG∗ (x) ∩ X | <  x∈V (H ) |NG (x) ∩ V (H )|. Combining this with (4), we obtain

     k (n − k) k − ≤ |NG∗ (x) ∩ X | < |NG (x) ∩ V (H )|. p

x∈V (H )

x∈Y

This implies that there is a vertex u0 ∈ V (H ) such that

|NG (u ) ∩ V (H )| ≥ k − 0

  k

p

+ 1.

(5)

Case 1. k ≤ p. In this case, ⌈ pk ⌉ = 1 and so |NG (u0 ) ∩ V (H )| ≥ k. Since |E (H )| < |E (H ∗ )|, we have that H is not complete, which implies

that there exists one vertex v ∈ V (H ) such that |NG (v) ∩ V (H )| ≤ k − 2. Thus, H ′ = G[(V (H )\{v}) ∪ {u0 }] is connected and |E (H ′ )| > |E (H )|, a contradiction completing the proof of (a). Case 2. k > p ≥ 3. We first prove the claim that H − v is connected for any v ∈ V (H ). Suppose, on the contrary, that there exists a vertex u ∈ V (H ) such that H − u is not connected. Let H1 be a component of H − u with |V (H1 )| ≤ k−2 1 .

If V (H1 ) = {w}, then |NG (w) ∩ V (H )| = 1 and H − w is connected. Since k ≥ 4, we have that k − ⌈ pk ⌉ > 1. Therefore, by

(5), H ′ = G[(V (H )\{w}) ∪ {u0 }] is connected and contains at least |E (H )| − 1 + (k − ⌈ pk ⌉) > |E (H )| edges, a contradiction. Suppose that |V (H1 )| ≥ 2. Let T1 be a spanning tree of H1 and let w1 , w2 be two leaves of T1 . Then H1 − wi is connected for i = 1, 2. This implies that G[V (H1 )∪{u}]−w1 or G[V (H1 )∪{u}]−w2 is connected, and so H −w1 or H −w2 is connected. Without loss of generality, assume that H − w1 is connected. Since H1 is a component of H − u with |V (H1 )| ≤ k−2 1 , we have |NG (w1 ) ∩ V (H )| ≤ k−2 1 . Since p ≥ 3, we have that k − ⌈ pk ⌉ > k−2 1 . Therefore, by (5), H ′ = G[(V (H )\{w1 }) ∪ {u0 }] is

connected and contains at least |E (H )| − k−2 1 + (k − ⌈ pk ⌉) > |E (H )| edges, a contradiction again. The proof of this claim is complete. Now, we prove (b). Clearly, n ≥ k. The case n = k holds trivially, so we assume that n ≥ k + 1 and proceed by induction on n. Let v be a vertex in G. If v ̸∈ V (H ), then G − v contains a connected subgraph H of order k. If v ∈ V (H ), then, by the above claim, H − v is connected. Recalling that k − ⌈ pk ⌉ > 1 and |NG (u0 ) ∩ V (H )| ≥ k − ⌈ pk ⌉ + 1, we have that G − v contains a connected subgraph G[(V (H )\{v}) ∪ {u0 }] of order k. Let H ′ be a connected subgraph of G − v with order k which contains as many edges as possible. Then H ′ is also a connected subgraph of G with order k. This implies that |E (H ′ )| ≤ |E (H )|. Combining this with the induction hypothesis, we have that for any v ∈ V (G),

|E (G − v)| ≤ |E (Tp,n−1 )| + |E (H )| − |E (Tp,k )|.

(6)

Suppose that dG (v) > |E (Tp,n )| − |E (Tp,n−1 )| for any v ∈ V (G). Set n = ps + r, where 0 ≤ r < p. Then dG (v) > n − s − 1

if r ≥ 1 and dG (v) > n − s if r = 0. Therefore, if r ≥ 1, then |E (G)| ≥ n(n−s+1) G 2 with dG

n(n−s) 2 Tp,n

n(n−s) 2

>

n(n−s)−r (s+1) 2

= |E (Tp,n )|, contradicting

Theorem 2.1. If r = 0, then |E ( )| ≥ > = |E (Tp,n )|, contradicting Theorem 2.1 again. If there is a vertex v ∈ V (G) (v) ≤ |E ( )| − |E (Tp,n−1 )|, then, by (6), we have that |E (G)| = |E (G − v)| + dG (v) ≤ |E (Tp,n−1 )| + |E (H )| − |E (Tp,k )| + dG (v) ≤ |E (Tp,n )| + |E (H )| − |E (Tp,k )|. This completes the proof of (b).  In the following, we give a lower bound on the cardinality of the λk -fragments of a (p + 1)-clique-free graph G with

λk (G) < ξk (G).

Theorem 2.3. Let G be a λk -connected and (p + 1)-clique-free graph with λk (G) < ξk (G). 1 (a) If k ≤ p and rk (G) < p, then rk (G) ≥ max{k + 1, p−1 δ(G) − k + p− }. 1 p (b) If k ≤ p and rk (G) ≥ p, then rk (G) ≥ max{k + 1, p−1 δ(G) − k − 1}. p

(c) If k > p ≥ 3, then rk (G) ≥ max{k + 1, p−1 δ(G) − k − 1}. p

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S. Wang et al. / Discrete Applied Mathematics 181 (2015) 255–259

Proof. Let X be a λk -atom of G. If |X | = k, then ξk (G) ≤ |[X , X¯ ]| = λk (G), a contradiction. Therefore, rk (G) = |X | ≥ k + 1.  Let H be a maximum connected subgraph of G[X ] with order k. Since λk (G) = |[X , X¯ ]| = u∈X dG (u) − 2|E (G[X ])| and

λk (G) < ξk (G) ≤ |[V (H ), V (H )]| = 



u∈V (H )

dG (u) − 2|E (H )|, we can deduce that

dG (u) − 2|E (G[X ])| + 2|E (H )| < 0.

(7)

u∈X \V (H )

(a) Assume that k ≤ p and rk (G) < p. Then |E (G[X ])| ≤



(rk (G)−1)rk (G) 2

+ |E (H )| −

(k−1)k 2

. By (7), we have that

dG (u) − (rk (G) − 1)rk (G) + (k − 1)k < 0.

u∈X \V (H )

Since dG (u) ≥ δ(G) for any u ∈ X \V (H ), we have 0 > u∈X \V (H ) dG (u) − (rk (G) − 1)rk (G) + (k − 1)k ≥ (rk (G) − k)(δ(G) − rk (G) − k + 1). Combining this with rk (G) ≥ k + 1, it follows that



rk (G) − δ(G) + k − 1 ≥ 1. This, together with rk (G) < p and k ≤ p, implies

(p−1)rk (G) p

−δ(G)+ (p−p1)k + 2pp−1 −2 ≥ 0, that is, rk (G) ≥

(b) Assume that k ≤ p and rk (G) ≥ p. By Lemma 2.2 (a), |E (G[X ])| ≤ |E (Tp,rk (G) )| + |E (H )| − we have that



(k−1)k 2

p p−1

δ(G)−k+ p−1 1 .

. Combining this with (7),

dG (u) − 2|E (Tp,rk (G) )| + (k − 1)k < 0.

u∈X \V (H )

By (1) and the assumption rk (G) ≥ p, we get



0 >

dG (u) − 2|E (Tp,rk (G) )| + (k − 1)k

u∈X \V (H )

≥ δ(G)(rk (G) − k) −

(p − 1)rk2 (G) p

(p − 1)rk2 (G)

+

(p − 1)k2 p

−

(p − 1)k2

k(p − k) p k(rk (G) − k)

≥ δ(G)(rk (G) − k) − + − p p p   (p − 1)rk (G) (p − 1)k k − − . = (rk (G) − k) δ(G) − p

p

p

(p−1)rk (G)

− δ(G) + (p−p1)k + 1 > 0, that is, (p − 1)rk (G) > p pδ(G) − (p − 1)k − p. Therefore, (p − 1)rk (G) ≥ pδ(G) − (p − 1)k − p + 1, or equivalently rk (G) ≥ p−1 δ(G) − k − 1. (c) Assume that k > p ≥ 3. Set rk (G) = ps0 + r0 and k = ps1 + r1 , where 0 ≤ r0 , r1 < p. Then s0 ≥ s1 . This, together with rk (G) > k and p ≥ k, implies

p

By Lemma 2.2 (b), |E (G[X ])| ≤ |E (Tp,rk (G) )| + |E (H )| − |E (Tp,k )|. Combining this with (7), we have that



dG (u) − 2|E (Tp,rk (G) )| + 2|E (Tp,k )| < 0.

u∈X \V (H )

Applying (1) to this inequality, we get 0 >



dG (u) − (rk2 (G) − s0 rk (G) − r0 − r0 s0 ) + (k2 − s1 k − r1 − r1 s1 )

u∈X \V (H )

≥ δ(G)(rk (G) − k) − (rk2 (G) − k2 ) + (s0 rk (G) − s1 k) + (r0 − r1 ) + (r0 s0 − r1 s1 ) p−1 2 1 1 = δ(G)(rk (G) − k) − (rk (G) − k2 ) − (r02 − pr0 ) + (r12 − pr1 ) p

= δ(G)(rk (G) − k) −

p−1 p

p

( (G) − k ) + (r0 − r1 ) 1 − rk2

We first prove the claim that (r0 − r1 )(1 −

2

r0 + r1 p



.

(8)

) ≥ −(rk (G) − k). If r0 ≥ r1 and 1 − r0 +p r1 ≥ 0, then (r0 − r1 )(1 − r0 +p r1 ) ≥ ≤ 0, then (r0 − r1 )(1 − r0 +p r1 ) ≥ −(rk (G) − k).

r0 +r1 p r0 +r1 p

0 ≥ −(rk (G) − k). Similarly, if r0 ≤ r1 and 1 − Suppose that r0 > r1 and 1 −

p



< 0. Since 0 ≤ r0 , r1 < p, we have −1 < 1 − r0 +p r1 . Note that (rk (G) − k) − (r0 − r1 ) = (s0 − s1 )p ≥ 0. Hence (r0 − r1 )(1 − r0 +p r1 ) ≥ (rk (G) − k)(1 − r0 +p r1 ) ≥ −(rk (G) − k). r0 +r1 p

S. Wang et al. / Discrete Applied Mathematics 181 (2015) 255–259

Suppose that r0 < r1 and 1 − s0 > s1 . Therefore,

r0 +r1 p

259

> 0. Then 0 < rk (G) − k = p(s0 − s1 ) + (r0 − r1 ) < p(s0 − s1 ), which implies that

rk (G) − k = p(s0 − s1 ) + (r0 − r1 )

≥ p − (r0 + r1 )   r0 + r1 = p 1− p   r0 + r1 > (r1 − r0 ) 1 − . p

The proof of this claim is complete. p−1 Combining this claim with (8), we have that 0 > δ(G)(rk (G) − k) − p (rk2 (G) − k2 ) − (rk (G) − k) = (rk (G) − k)(δ(G) −

(G)− p−p 1 k−1). Since rk (G) > k, it follows that δ(G)− p−p 1 rk (G)− p−p 1 k−1 < 0. So, (p−1)rk (G) ≥ pδ(G)−(p−1)k−p+1, p or equivalently rk (G) ≥ p−1 δ(G) − k − 1.  p−1 rk p

By Theorems 1.2 and 2.3, we have the following result. Theorem 2.4. Let G be a λk -connected and (p + 1)-clique-free graph, where p ≥ 2. If λk (G) < ξk (G), then rk (G) ≥ max{k + 1, p δ(G) − k − 1}. p−1 Corollary 2.5. Let G be a λk -connected and (p + 1)-clique-free graph with λk (G) ≤ ξk (G), where p ≥ 2. If δ(G) ≥ (rk (G) + k) + 1, then G is λk -optimal.

p−1 p

Proof. Assume to the contrary that G is not λk -optimal. Then λk (G) < ξk (G). By Theorem 2.4, rk (G) ≥ p−1 δ(G) − k − 1, that p−1 p−1 p−1 is, δ(G) ≤ p (rk (G) + k) + p . This contradicts the assumption that δ(G) ≥ p (rk (G) + k) + 1.  p

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