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On maximal 3-restricted edge connectivity and reliability analysis of hypercube networks
Applied Mathematics and Computation 217 (2010) 2602–2607
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
On maximal 3-restricted edge connectivity and reliability analysis ofhypercube networks Jianping Ou Department of Mathematics & Physics, Wuyi University, Jiangmen 529020, China
a r t i c l e
i n f o
a b s t r a c t It is shown in this work that all n-dimensional hypercube networks for n P 4 are maximally 3-restricted edge connected. Employing this observation, we analyze the reliability of hypercube networks and determine the first 3n 5 coefficients of the reliability polynomial of n-cube networks. Ó 2010 Published by Elsevier Inc.
Keywords: Hypercube network Edge connectivity Network reliability
1. Introduction When modelling reliability of telecommunication networks, it is reasonable to assume that their nodes (vertices) never fail but edges (links) fail independently of each other with equal probability p < 1. This kind of network models are often called Moore–Shannon ones [1,2]. Let M be a Moore–Shannon network model that has size e (the number of its edges) and edge failure probability p < 1. If denote by Ch the number of its edge cuts that have size h, then the reliability of M, namely the probability it remains connectedness, can be expressed as:
RðM; pÞ ¼ 1
e X
C h ph ðl pÞeh :
h¼1
The right hand side is called its reliability polynomial. To determine the reliability, one must calculate all the coefficients Ch. But, Provan proves in [3] that it is NP-hard to calculate all these coefficients. With super restricted edge connectivity, Bauer gives a general expression of those coefficients Ch with h 6 k(M) in [4], where k(M) is the edge connectivity of M. In order to estimate more precisely the reliability, Esfahanian introduces the concepts of restricted edge cut and restricted edge connectivity in [5]. These concepts are generalized in [6–8] as follows. Definition 1.1. An m-restricted edge cut is an edge cut of a connected graph which disconnects this graph with each component having order at least m. The size of a minimum m-restricted edge cut of a graph G is called its m-restricted edge connectivity. We denote by km(G), or simply km, the m-restricted edge connectivity of graph G in this paper. 2-restricted edge cuts and 2-restricted edge connectivity are the so-called restricted edge cuts and restricted edge connectivity introduced by Esfahanian in [5]. Employing the properties of m-restricted edge cuts and m-restricted edge connectivity with m 6 3, Li in [9] and Wang in [10] determine the first k3 1 coefficients of the reliability polynomial of circulant networks. Their results show that for Moore–Shannon models with same number of nodes and size those that have larger m-restricted edge connectivity for all m 6 3 are locally more reliable, where a network M is called locally more reliable than another network N if there exists a positive integer p0 < 1 such that R(M, p) > R(N, p) holds for all positive integers p 6 p0. And so, graphs with maximal 3-restricted edge connectivity are important in the design of locally most reliable networks. E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2010 Published by Elsevier Inc. doi:10.1016/j.amc.2010.07.073
J. Ou / Applied Mathematics and Computation 217 (2010) 2602–2607
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For advances on m-restricted edge connectivity, the readers are suggested to refer to [6,8,11–14]. Graphs that are edgetransitive and vertex-transitive are maximally 3-restricted edge connected [15]. This property is also enjoyed by vertex-transitive graph of girth at least 5, refer to [10] for example. For details on 3-restricted edge connectivity of vertex-transitive graphs, the readers are suggested to refer to [16]. In general, undirected binary Kautz graphs are maximally 3-restricted edge connected [14], so does the minimally 3-restricted edge connected graphs [13]. For recent advances on maximizing 3-restricted edge connectivity of graphs, the readers are suggested to refer to a survey [12]. Conditions for optimizing m-restricted edge connectivity of special networks are obtained in [14,17] and elsewhere. In this work, it is shown that in general, hypercube networks are maximally 3-restricted edge connected. This observation enable us to present an explicit expression of Ch for these networks, where k(M) 6 h 6 k3(M) 1. We remark here that a hypercube graph G is an n-cube with n P 4, it has vertex set V(G) = {(u1, u2, . . . , un): ui 2 {0, 1}}, where a vertex (u1, u2, . . . , un) is adjacent to another vertex (v1, v2, . . . , vn) if and only if there is a unique j 2 {1, 2, . . . , n} such that uj – vj. Our main results are Theorem 3.2. An n-cube is maximally 3-restricted edge connected if n P 4. Theorem 4.2. Let G be an n cube with size e and Ch be the number of its edge cuts of size h. If n P 4 then
8 0; if h < n > > > > en > n > > ; if n 6 h 6 2n 3 >2 > hn > > > > > > < 2n e n þ e e 2n þ 1 hn h 2n þ 2 Ch ¼ > n > > e 2n þ 1 2 > > > e e > > h 2n þ 1 2 > > > > > e 2n > > : ; if 2n 2 6 h 6 3n 5 h 2n Before proceeding, let us introduce some more notations and terminologies. For a subgraph X of graph G or a subset of V(G), GnX indicates the graph obtained by removing all the vertices of X from G. Let G[X] represent the subgraph of G induced by the vertex set of X. If B is another subgraph or subset disjoint with X, then [X, B] indicates the set of edges of G with one endpoint in X and the other in B. We simplify [X, GnX] as I(X), and [{u}, X] as [u, X], where u R X or V(X). Let @(X) = jI(X)j, nm(G) = min{@(X): X is a connected vertex-induced subgraph of order m of graph G}. Then k3(G) 6 n3(G) [4], graph G is called maximally 3-restricted edge connected if the equality holds. Let g(G) represent the girth of graph G, namely the length of its shortest cycles. Denote by m(G) or jGj the order of graph G, and e(G) its size (the number of its edges). For other symbols and terminology not specified, we follow that of [18].
2. Properties of fragment In this section, we study the properties of 3-restricted fragments. For any minimum 3-restricted edge cut S of graph G, G S consists of two components. We call these two components 3-restricted fragments, or simply fragments, of G corresponding to S, between which the smaller one (with less vertices) is called a normal fragment. Among all fragments, those that have minimum cardinality are called atoms. We often signify a fragment with its vertex set, and wish that this lead to no confusion. It is worth noting that a fragment referred here is a connected vertex-induced subgraph corresponding to some minimum 3-restricted edge cut. They appear in pairs and are essentially different from the traditional ones. If denote by X one fragment corresponding to some minimum 3-restricted edge cut S, then the other one is denoted by Xc. For conciseness, we restrict our attention to vertex-transitive graphs of girth 4 and degree k > 3 in the following discussion. A graph G is vertex-transitive if, for any two vertices u and v of G, there is an automorphism s 2 Aut(G) such that s(u) = v, where Aut(G) is the automorphism group of graph G. Clearly, a vertex-transitive graph is k-regular for some integer k. Since every connected subgraph of order 3 of a k-regular vertex-transitive graph G of girth 4 is a path, it follows that n3(G) = 3k 4. Lemma 2.1 [19]. A k-regular connected vertex-transitive graph is k-edge connected. Lemma 2.2 [20]. Let G be a triangle-free k-regular connected vertex transitive graph with k P 3. Then k2(G) = 2k 2. Lemma 2.3. Let G be a k-regular connected graph of girth 4 and order at least 6. If k P 3, then k2(G) 6 k3(G) 6 n3(G) = 3k 4. Proof. Since every 3-restricted edge cut is a 2-restricted edge cut, the first inequality is obviously true. Let P be a connected vertex-induced subgraph of order 3. Then P is a path. If GnP contains a component F with jFj 6 2, then G[F [ P] contains a 3-
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cycle, which is impossible. Hence, every component of GnP has order at least three. It follows that I(P) is a 3-restricted edge cut of size n3(G), and so k3(G) 6 @(P) = n3(G) as desired. h Lemma 2.4. Let G be a k-regular connected vertex-transitive graph of girth 4 with k P 3, X and Y be two of its distinct fragments. If the following two conditions hold, then X \ Y is a fragment.
ð1Þ jX \ Yj > 3;
ð2Þ @ðX \ YÞ < k3 ðGÞ:
Proof. Let X and Xc be the two fragments corresponding to a minimum 3-restricted edge cut S, Y and Yc be the two fragments corresponding to another minimum 3-restricted edge cut T. Define
A ¼ X \ Y;
B ¼ X \ YC;
C ¼ X C \ Y;
D ¼ XC \ Y C :
Since Xc GnA and jGnAj P 3, it suffices to show that both A and Ac = GnA are connected. Since Xc and Yc are two connected fragments, Ac is connected whenever D – ;. When D = ;, if [B, C] = ; then S = [A, C], T = [A, B] and I(A) = S [ T. Therefore
@ðAÞ ¼ jSj þ jTj ¼ 2k3 ðGÞ > k3 ðGÞ: This contradiction implies that [B, C] – ; when D = ;. Hence, Ac is connected in either case. Suppose, to the contrary, that A is not connected with components Ai, i = 1, 2, . . . , m, m P 2. Then m X
@ðAi Þ ¼ @ðAÞ 6 k3 ðGÞ 6 3k 4:
i¼1
If m P 3 then @(Aj) 6 k 1 for some Aj, which contradicts Lemma 2.1. Hence m = 2. By Condition (1), A has a component, say A1, of order at least two. It follows that I(A1) is a 2-restricted edge cut. From Lemma 2.2, we have @(A1) P 2k 2. Hence
@ðA2 Þ ¼ @ðAÞ @ðA1 Þ 6 3k 4 ð2k 2Þ ¼ k 2: This contradicts Lemma 2.1. And so, A is connected. h Lemma 2.5. Let G be a k-regular connected vertex-transitive graph of girth 4 with k P 3, X and Y be two of its distinct normal fragments. If jX \ Yj P 3, then X \ Y is a fragment. Proof. By Lemma 2.4, it suffices to show that @(X \ Y) 6 k3(G). Let X and Y be the two normal fragments corresponding to minimum 3-restricted edge cut S and T respectively. Define A, B, C and D the same as in the proof of Lemma 2.4. Since jAj + jBj = jXj 6 Gj/2 6 Ycj = jBj + jDj, we have
jDj P jAj ¼ jX \ Yj P 3:
ð1Þ
If D is connected, by Formula (1), I(D) is a 3-restricted edge cut, and so @(D) P k3(G); if D has at least three components, by Lemma 2.1, @(D) P 3k(G) = 3k > k3(G); if D contains exactly two components, then one of these two components has order at least 2. From Lemmas 2.1 and 2.2, we deduce that @(D) P k2(G) + k(G) = 3k 2 > k3(G). These observations show that @(D) P k3(G), the equality holds if and only if D is a fragment. Therefore
@ðAÞ þ @ðDÞ ¼ j½A; Bj þ j½A; Cj þ 2j½A; Dj þ j½B; Dj þ j½D; Cj þ 2j½B; Cj 2j½B; Cj ¼ jSj þ jTj 2j½B; Cj 6 jSj þ jTj ¼ 2k3 ðGÞ: It follows from the above formula that @(X \ Y) = @(A) 6 k3(G).
h
Noticing that atoms are normal fragments that contain no fragments as their proper subgraphs, according to Lemma 2.5 we have the following Lemma 2.6. Let G be a k-regular connected vertex-transitive graph of girth 4 with k P 3. If X and Y are two distinct atoms of G, then jX \ Yj 6 2. Lemma 2.7. Let G be a k-regular connected vertex-transitive graph of girth 4 with k P 3. If G is not maximally 3-restricted edge connected, then its atoms are vertex-disjoint unless k = 3. Proof. Suppose, to the contrary, that graph G contains two atoms X and Y such that X \ Y – ; when k > 3. Define A, B, C and D the same as in the proof of Lemma 2.4. Since G is not maximally 3-restricted edge connected, it follows that k3(G) 6 3k 5. Let H be a connected vertex-induced subgraph. Since g(G) = 4, @(H) = 3k 4 if jHj = 3 and @(H) 6 4k 8 if jHj = 4. In either case, we have @(H) P 3k 5 P k3(G). And so, H cannot be a fragment when jHj 6 4 unless k = 3, k3(G) = 3k 5 and H is a 4-cycle. This observation shows that jXj = jYj P 5 when k P 4. By Lemma 2.6, jBj = jCj P 3 unless k = 3. On the other hand, we have @(B) + @(C)6@(X) + @(Y) = 2k3(G). Therefore
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@ðBÞ 6 k3 ðGÞ or @ðGÞ 6 k3 ðGÞ: It follows from Lemma 2.4 that either B or C is a fragment. But they are proper subgraphs of an atom X or Y. Lemma 2.7 follows from this contradiction. h Lemma 2.8. Let G be a k-regular connected vertex-transitive graph of girth 4 with k P 3, and X be one of its atom. If k3(G) 6 3k 5, then X is vertex-transitive unless k = 3. Proof. Let u and v be two arbitrary vertices of X. Since G is vertex-transitive, there is an automorphism s 2 Aut(G) such that s(u) = v. Note that s(X) is also an atom with s(X) \ X – ;. By Lemma 2.7, s(X) = X unless k = 3. Since the restriction of s on X is an automorphism of X, it follows that X is also vertex-transitive unless k = 3. h Lemma 2.9. Let X be a fragment of a k-regular triangle-free connected graph G. If G is not maximally 3-restricted edge connected, then jXj 6 2k 2. Proof. Since G is triangle-free, e(X) 6 jXj2/4. Let dX(u) indicate the degree of vertex u in X. Combining the previous inequality with @(X) = k3(G) 6 3k 5, we have
kjXj ð3k 5Þ 6 kjXj k3 ðGÞ ¼ kjXj @ðXÞ ¼
X
dX ðuÞ ¼ 2eðXÞ 6 jXj2 =2:
u2X
Hence
ðjXj 3ÞðjXj ð2k 3ÞÞ > 0: Since G is not maximally 3-restricted edge connected, it follows that jXj 3 > 0. And so jXj P 2k 2.
h
3. Restricted edge connectivity
Lemma 3.1. Let G be a connected k-regular vertex-transitive graph with g(G) = 4, k P 4 and jGj P 6. If G is not maximally 3restricted edge connected, then k3(G) is a divisor of jGj such that 2k 2 6 k3(G) 6 3k 5. Furthermore, the 3-restricted atom is (k 1)-regular vertex-transitive graph of order k3(G). Proof. Let X be a 3-restricted atom of G. Since G is not maximal 3-restricted edge connected, by Lemma 2.8, X is an r-regular vertex-transitive graph with 0 < r < k. From Lemma 2.9, we deduce that jXj P 2k 2. Therefore
ðk rÞð2k 2Þ 6 ðk rÞjXj ¼ kjXj rjXj ¼ @ðXÞ ¼ @ 3 ðGÞ 6 3k 5: Combining this observation with k > 3 and 0 < r < k, we obtain r = k 1 and 2k 2 6 jXj = k3(G) 6 3k 5. The lemma follows. h Theorem 3.2. An n-cube is maximally 3-restricted edge connected if n P 4. Proof. Let G be an n-cube with vertex set V(G) = {(u1, u2, . . . , un): ui = 0 or 1, n P 4}. Then it is an n-regular vertex-transitive graph with g(G) = 4 and n P 4, two distinct vertices (u1, u2, . . . , un) and (v1, v2, . . . , vn) have a common neighbor (w1, w2, . . . , wn) if and only is there are exactly two integers j, k 2 {1, 2, . . . , n} such that uj – wj, vk – ek and j – k. It follows that for any two vertices x, y 2 V(G), jN(x)C \ N(y)j = 0 or 2. Let X be an arbitrary 3-restricted atom. If G is not maximally 3-restricted edge connected, namely k3(G) 6 3n 5, then X is an (n 1)-regular vertex-transitive graph with 2n 2 6 jXj 6 3n 5. For any given vertex u 2 X, since n P 4, the neighborhood NX(u) of vertex u in X contains at least three vertices, say, x, y, z. Noticing that no neighbors of a given vertex are adjacent to one another, and that jN(x) \ N(y)j = 0 or 2, we have jNX(x) [ NX(y)j P 2(n 1) 2 = 2n 4. Since, in X, vertex z and x (or y) have at most one common neighbor other than u, it follows that
jNX ðxÞ [ NX ðyÞ [ N X ðzÞj P ð2n 4Þ þ ðn 1 2Þ ¼ 3n 7: Therefore jXj P jNX(x) [ NX(y) [ NX(z)j + jNX(u)j P 4n 8. By the second part of Lemma 3.1, we have n3(G) = jXj P 4n 8 P 3n 4. This contradicts our hypothesis, and so G is maximally 3-restricted edge connected. h 4. Reliability analysis Let G be an n-cube with n P 4. Then it is an n-regular connected vertex-transitive graph with order 2n, size e = n 2n1 and girth 4. From Lemmas 2.1 and 2.2, and Theorem 3.2, we deduce that k(G) = n, k2(G) = 2n 2 and k3(G) = 3n 4. Let S be
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an edge cut of size h. Then e P h P n. If 2n 3 P h P n then G S consists of two components, one of which is an isolated vertex; if 3n 5 P h P 2n 2 then either G S consists of three components, two of which are isolated vertices, or G S consists of two components, one of which is an isolated vertex or isolated edge. Let E(u) denote the set of edge cuts of size at most 3n 5 of graph G that separates vertex u from G, and E(f) be the set of edge cuts of size at most 3n 5 of graph G that separates edge f from G. With previous discussion and symbols, we have: Lemma 4.1. Let u be a vertex of n-cube G and f be one of its edges. If n > 4 then n If define ¼ 0 when m < 0, we have the following result. m
S
u2VðGÞ EðuÞ
\
S
f 2EðGÞ Eðf Þ
¼ ;.
Theorem 4.2. Let G be an n-cube with size e and Ch be the number of its edge cuts of size h. If n > 4 then
8 0; if h < n > > > > e n > > > 2n ; if 2n 3 P h P n > > hn > > > > > > < 2n e n þ e e 2n þ 1 hn h 2n þ 2 Ch ¼ > n > > e 2n þ 1 2 > > > e 2 > > h 2n þ 2 2 > > > > > e 2n > > : ; if 3n 5 P h P 2n 2 h 2n Proof. From Lemma 2.1 it follows that Ch = 0 when h < n. Let S be an edge cut of size h. If 2n 3 > h > n then G S consists of two components, one of which is an isolated vertex. Let Ch(u) denote the number of edge cuts of size h that separate vertex u from G. Then
C h ðuÞ ¼
en hn
It follows that
C h ¼ 2n
:
en hn
:
If 3n 5 > h > 2n 2, then the set of edge cuts of size h consists of two disjoint subsets by Lemma 4.1 and its proof. The first one formed by those edge cuts that separate one or two vertices from G. This set has cardinality
[ X X EðuÞ ¼ C ðuÞ jEðuÞ \ Eðv Þj: u2VðGÞ h u2VðGÞ u;v 2VðGÞ
If vertex u and
v are adjacent to each other, then
jEðuÞ \ Eðv Þj ¼
e 2n þ 1 h 2n þ 1
:
If vertex u is not adjacent to v, then
jEðuÞ \ Eðv Þj ¼
e 2n h 2n
:
Therefore
[ en e 2n þ 1 2n e 2n EðuÞ ¼ 2n e e : u2VðGÞ hn h 2n þ 1 2 h 2n
ð2Þ
Each of the second kind edge cuts separates an isolated edge from G. Hence, the number of this kind edge cuts is
e
e 2n þ 1 h 2n þ 2
:
The second part of the theorem follows from the combination of (2) and (3).
ð3Þ h
5. Conclusion Let G be an n-cube with n > 4. If its nodes do not fail and edges fail independently with equal probability p < 1, then the coefficients Ch in its reliability polynomial can be calculated according to Theorem 4.2 when 3n5 > h > 2n 2, and Ch = 0 when h < n. Hence, an approximation of precision O(p3n-4) of R(G, p) is derived when p ? 0.
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Acknowledgement Supported by National Natural Science Foundation of China (Grant No. 10801091); NSF of Guandong Province (Grant Nos. 8152902001000004; 9151051501000072). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
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Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
On maximal 3-restricted edge connectivity and reliability analysis ofhypercube networks Jianping Ou Department of Mathematics & Physics, Wuyi University, Jiangmen 529020, China
a r t i c l e
i n f o
a b s t r a c t It is shown in this work that all n-dimensional hypercube networks for n P 4 are maximally 3-restricted edge connected. Employing this observation, we analyze the reliability of hypercube networks and determine the first 3n 5 coefficients of the reliability polynomial of n-cube networks. Ó 2010 Published by Elsevier Inc.
Keywords: Hypercube network Edge connectivity Network reliability
1. Introduction When modelling reliability of telecommunication networks, it is reasonable to assume that their nodes (vertices) never fail but edges (links) fail independently of each other with equal probability p < 1. This kind of network models are often called Moore–Shannon ones [1,2]. Let M be a Moore–Shannon network model that has size e (the number of its edges) and edge failure probability p < 1. If denote by Ch the number of its edge cuts that have size h, then the reliability of M, namely the probability it remains connectedness, can be expressed as:
RðM; pÞ ¼ 1
e X
C h ph ðl pÞeh :
h¼1
The right hand side is called its reliability polynomial. To determine the reliability, one must calculate all the coefficients Ch. But, Provan proves in [3] that it is NP-hard to calculate all these coefficients. With super restricted edge connectivity, Bauer gives a general expression of those coefficients Ch with h 6 k(M) in [4], where k(M) is the edge connectivity of M. In order to estimate more precisely the reliability, Esfahanian introduces the concepts of restricted edge cut and restricted edge connectivity in [5]. These concepts are generalized in [6–8] as follows. Definition 1.1. An m-restricted edge cut is an edge cut of a connected graph which disconnects this graph with each component having order at least m. The size of a minimum m-restricted edge cut of a graph G is called its m-restricted edge connectivity. We denote by km(G), or simply km, the m-restricted edge connectivity of graph G in this paper. 2-restricted edge cuts and 2-restricted edge connectivity are the so-called restricted edge cuts and restricted edge connectivity introduced by Esfahanian in [5]. Employing the properties of m-restricted edge cuts and m-restricted edge connectivity with m 6 3, Li in [9] and Wang in [10] determine the first k3 1 coefficients of the reliability polynomial of circulant networks. Their results show that for Moore–Shannon models with same number of nodes and size those that have larger m-restricted edge connectivity for all m 6 3 are locally more reliable, where a network M is called locally more reliable than another network N if there exists a positive integer p0 < 1 such that R(M, p) > R(N, p) holds for all positive integers p 6 p0. And so, graphs with maximal 3-restricted edge connectivity are important in the design of locally most reliable networks. E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2010 Published by Elsevier Inc. doi:10.1016/j.amc.2010.07.073
J. Ou / Applied Mathematics and Computation 217 (2010) 2602–2607
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For advances on m-restricted edge connectivity, the readers are suggested to refer to [6,8,11–14]. Graphs that are edgetransitive and vertex-transitive are maximally 3-restricted edge connected [15]. This property is also enjoyed by vertex-transitive graph of girth at least 5, refer to [10] for example. For details on 3-restricted edge connectivity of vertex-transitive graphs, the readers are suggested to refer to [16]. In general, undirected binary Kautz graphs are maximally 3-restricted edge connected [14], so does the minimally 3-restricted edge connected graphs [13]. For recent advances on maximizing 3-restricted edge connectivity of graphs, the readers are suggested to refer to a survey [12]. Conditions for optimizing m-restricted edge connectivity of special networks are obtained in [14,17] and elsewhere. In this work, it is shown that in general, hypercube networks are maximally 3-restricted edge connected. This observation enable us to present an explicit expression of Ch for these networks, where k(M) 6 h 6 k3(M) 1. We remark here that a hypercube graph G is an n-cube with n P 4, it has vertex set V(G) = {(u1, u2, . . . , un): ui 2 {0, 1}}, where a vertex (u1, u2, . . . , un) is adjacent to another vertex (v1, v2, . . . , vn) if and only if there is a unique j 2 {1, 2, . . . , n} such that uj – vj. Our main results are Theorem 3.2. An n-cube is maximally 3-restricted edge connected if n P 4. Theorem 4.2. Let G be an n cube with size e and Ch be the number of its edge cuts of size h. If n P 4 then
8 0; if h < n > > > > en > n > > ; if n 6 h 6 2n 3 >2 > hn > > > > > > < 2n e n þ e e 2n þ 1 hn h 2n þ 2 Ch ¼ > n > > e 2n þ 1 2 > > > e e > > h 2n þ 1 2 > > > > > e 2n > > : ; if 2n 2 6 h 6 3n 5 h 2n Before proceeding, let us introduce some more notations and terminologies. For a subgraph X of graph G or a subset of V(G), GnX indicates the graph obtained by removing all the vertices of X from G. Let G[X] represent the subgraph of G induced by the vertex set of X. If B is another subgraph or subset disjoint with X, then [X, B] indicates the set of edges of G with one endpoint in X and the other in B. We simplify [X, GnX] as I(X), and [{u}, X] as [u, X], where u R X or V(X). Let @(X) = jI(X)j, nm(G) = min{@(X): X is a connected vertex-induced subgraph of order m of graph G}. Then k3(G) 6 n3(G) [4], graph G is called maximally 3-restricted edge connected if the equality holds. Let g(G) represent the girth of graph G, namely the length of its shortest cycles. Denote by m(G) or jGj the order of graph G, and e(G) its size (the number of its edges). For other symbols and terminology not specified, we follow that of [18].
2. Properties of fragment In this section, we study the properties of 3-restricted fragments. For any minimum 3-restricted edge cut S of graph G, G S consists of two components. We call these two components 3-restricted fragments, or simply fragments, of G corresponding to S, between which the smaller one (with less vertices) is called a normal fragment. Among all fragments, those that have minimum cardinality are called atoms. We often signify a fragment with its vertex set, and wish that this lead to no confusion. It is worth noting that a fragment referred here is a connected vertex-induced subgraph corresponding to some minimum 3-restricted edge cut. They appear in pairs and are essentially different from the traditional ones. If denote by X one fragment corresponding to some minimum 3-restricted edge cut S, then the other one is denoted by Xc. For conciseness, we restrict our attention to vertex-transitive graphs of girth 4 and degree k > 3 in the following discussion. A graph G is vertex-transitive if, for any two vertices u and v of G, there is an automorphism s 2 Aut(G) such that s(u) = v, where Aut(G) is the automorphism group of graph G. Clearly, a vertex-transitive graph is k-regular for some integer k. Since every connected subgraph of order 3 of a k-regular vertex-transitive graph G of girth 4 is a path, it follows that n3(G) = 3k 4. Lemma 2.1 [19]. A k-regular connected vertex-transitive graph is k-edge connected. Lemma 2.2 [20]. Let G be a triangle-free k-regular connected vertex transitive graph with k P 3. Then k2(G) = 2k 2. Lemma 2.3. Let G be a k-regular connected graph of girth 4 and order at least 6. If k P 3, then k2(G) 6 k3(G) 6 n3(G) = 3k 4. Proof. Since every 3-restricted edge cut is a 2-restricted edge cut, the first inequality is obviously true. Let P be a connected vertex-induced subgraph of order 3. Then P is a path. If GnP contains a component F with jFj 6 2, then G[F [ P] contains a 3-
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cycle, which is impossible. Hence, every component of GnP has order at least three. It follows that I(P) is a 3-restricted edge cut of size n3(G), and so k3(G) 6 @(P) = n3(G) as desired. h Lemma 2.4. Let G be a k-regular connected vertex-transitive graph of girth 4 with k P 3, X and Y be two of its distinct fragments. If the following two conditions hold, then X \ Y is a fragment.
ð1Þ jX \ Yj > 3;
ð2Þ @ðX \ YÞ < k3 ðGÞ:
Proof. Let X and Xc be the two fragments corresponding to a minimum 3-restricted edge cut S, Y and Yc be the two fragments corresponding to another minimum 3-restricted edge cut T. Define
A ¼ X \ Y;
B ¼ X \ YC;
C ¼ X C \ Y;
D ¼ XC \ Y C :
Since Xc GnA and jGnAj P 3, it suffices to show that both A and Ac = GnA are connected. Since Xc and Yc are two connected fragments, Ac is connected whenever D – ;. When D = ;, if [B, C] = ; then S = [A, C], T = [A, B] and I(A) = S [ T. Therefore
@ðAÞ ¼ jSj þ jTj ¼ 2k3 ðGÞ > k3 ðGÞ: This contradiction implies that [B, C] – ; when D = ;. Hence, Ac is connected in either case. Suppose, to the contrary, that A is not connected with components Ai, i = 1, 2, . . . , m, m P 2. Then m X
@ðAi Þ ¼ @ðAÞ 6 k3 ðGÞ 6 3k 4:
i¼1
If m P 3 then @(Aj) 6 k 1 for some Aj, which contradicts Lemma 2.1. Hence m = 2. By Condition (1), A has a component, say A1, of order at least two. It follows that I(A1) is a 2-restricted edge cut. From Lemma 2.2, we have @(A1) P 2k 2. Hence
@ðA2 Þ ¼ @ðAÞ @ðA1 Þ 6 3k 4 ð2k 2Þ ¼ k 2: This contradicts Lemma 2.1. And so, A is connected. h Lemma 2.5. Let G be a k-regular connected vertex-transitive graph of girth 4 with k P 3, X and Y be two of its distinct normal fragments. If jX \ Yj P 3, then X \ Y is a fragment. Proof. By Lemma 2.4, it suffices to show that @(X \ Y) 6 k3(G). Let X and Y be the two normal fragments corresponding to minimum 3-restricted edge cut S and T respectively. Define A, B, C and D the same as in the proof of Lemma 2.4. Since jAj + jBj = jXj 6 Gj/2 6 Ycj = jBj + jDj, we have
jDj P jAj ¼ jX \ Yj P 3:
ð1Þ
If D is connected, by Formula (1), I(D) is a 3-restricted edge cut, and so @(D) P k3(G); if D has at least three components, by Lemma 2.1, @(D) P 3k(G) = 3k > k3(G); if D contains exactly two components, then one of these two components has order at least 2. From Lemmas 2.1 and 2.2, we deduce that @(D) P k2(G) + k(G) = 3k 2 > k3(G). These observations show that @(D) P k3(G), the equality holds if and only if D is a fragment. Therefore
@ðAÞ þ @ðDÞ ¼ j½A; Bj þ j½A; Cj þ 2j½A; Dj þ j½B; Dj þ j½D; Cj þ 2j½B; Cj 2j½B; Cj ¼ jSj þ jTj 2j½B; Cj 6 jSj þ jTj ¼ 2k3 ðGÞ: It follows from the above formula that @(X \ Y) = @(A) 6 k3(G).
h
Noticing that atoms are normal fragments that contain no fragments as their proper subgraphs, according to Lemma 2.5 we have the following Lemma 2.6. Let G be a k-regular connected vertex-transitive graph of girth 4 with k P 3. If X and Y are two distinct atoms of G, then jX \ Yj 6 2. Lemma 2.7. Let G be a k-regular connected vertex-transitive graph of girth 4 with k P 3. If G is not maximally 3-restricted edge connected, then its atoms are vertex-disjoint unless k = 3. Proof. Suppose, to the contrary, that graph G contains two atoms X and Y such that X \ Y – ; when k > 3. Define A, B, C and D the same as in the proof of Lemma 2.4. Since G is not maximally 3-restricted edge connected, it follows that k3(G) 6 3k 5. Let H be a connected vertex-induced subgraph. Since g(G) = 4, @(H) = 3k 4 if jHj = 3 and @(H) 6 4k 8 if jHj = 4. In either case, we have @(H) P 3k 5 P k3(G). And so, H cannot be a fragment when jHj 6 4 unless k = 3, k3(G) = 3k 5 and H is a 4-cycle. This observation shows that jXj = jYj P 5 when k P 4. By Lemma 2.6, jBj = jCj P 3 unless k = 3. On the other hand, we have @(B) + @(C)6@(X) + @(Y) = 2k3(G). Therefore
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@ðBÞ 6 k3 ðGÞ or @ðGÞ 6 k3 ðGÞ: It follows from Lemma 2.4 that either B or C is a fragment. But they are proper subgraphs of an atom X or Y. Lemma 2.7 follows from this contradiction. h Lemma 2.8. Let G be a k-regular connected vertex-transitive graph of girth 4 with k P 3, and X be one of its atom. If k3(G) 6 3k 5, then X is vertex-transitive unless k = 3. Proof. Let u and v be two arbitrary vertices of X. Since G is vertex-transitive, there is an automorphism s 2 Aut(G) such that s(u) = v. Note that s(X) is also an atom with s(X) \ X – ;. By Lemma 2.7, s(X) = X unless k = 3. Since the restriction of s on X is an automorphism of X, it follows that X is also vertex-transitive unless k = 3. h Lemma 2.9. Let X be a fragment of a k-regular triangle-free connected graph G. If G is not maximally 3-restricted edge connected, then jXj 6 2k 2. Proof. Since G is triangle-free, e(X) 6 jXj2/4. Let dX(u) indicate the degree of vertex u in X. Combining the previous inequality with @(X) = k3(G) 6 3k 5, we have
kjXj ð3k 5Þ 6 kjXj k3 ðGÞ ¼ kjXj @ðXÞ ¼
X
dX ðuÞ ¼ 2eðXÞ 6 jXj2 =2:
u2X
Hence
ðjXj 3ÞðjXj ð2k 3ÞÞ > 0: Since G is not maximally 3-restricted edge connected, it follows that jXj 3 > 0. And so jXj P 2k 2.
h
3. Restricted edge connectivity
Lemma 3.1. Let G be a connected k-regular vertex-transitive graph with g(G) = 4, k P 4 and jGj P 6. If G is not maximally 3restricted edge connected, then k3(G) is a divisor of jGj such that 2k 2 6 k3(G) 6 3k 5. Furthermore, the 3-restricted atom is (k 1)-regular vertex-transitive graph of order k3(G). Proof. Let X be a 3-restricted atom of G. Since G is not maximal 3-restricted edge connected, by Lemma 2.8, X is an r-regular vertex-transitive graph with 0 < r < k. From Lemma 2.9, we deduce that jXj P 2k 2. Therefore
ðk rÞð2k 2Þ 6 ðk rÞjXj ¼ kjXj rjXj ¼ @ðXÞ ¼ @ 3 ðGÞ 6 3k 5: Combining this observation with k > 3 and 0 < r < k, we obtain r = k 1 and 2k 2 6 jXj = k3(G) 6 3k 5. The lemma follows. h Theorem 3.2. An n-cube is maximally 3-restricted edge connected if n P 4. Proof. Let G be an n-cube with vertex set V(G) = {(u1, u2, . . . , un): ui = 0 or 1, n P 4}. Then it is an n-regular vertex-transitive graph with g(G) = 4 and n P 4, two distinct vertices (u1, u2, . . . , un) and (v1, v2, . . . , vn) have a common neighbor (w1, w2, . . . , wn) if and only is there are exactly two integers j, k 2 {1, 2, . . . , n} such that uj – wj, vk – ek and j – k. It follows that for any two vertices x, y 2 V(G), jN(x)C \ N(y)j = 0 or 2. Let X be an arbitrary 3-restricted atom. If G is not maximally 3-restricted edge connected, namely k3(G) 6 3n 5, then X is an (n 1)-regular vertex-transitive graph with 2n 2 6 jXj 6 3n 5. For any given vertex u 2 X, since n P 4, the neighborhood NX(u) of vertex u in X contains at least three vertices, say, x, y, z. Noticing that no neighbors of a given vertex are adjacent to one another, and that jN(x) \ N(y)j = 0 or 2, we have jNX(x) [ NX(y)j P 2(n 1) 2 = 2n 4. Since, in X, vertex z and x (or y) have at most one common neighbor other than u, it follows that
jNX ðxÞ [ NX ðyÞ [ N X ðzÞj P ð2n 4Þ þ ðn 1 2Þ ¼ 3n 7: Therefore jXj P jNX(x) [ NX(y) [ NX(z)j + jNX(u)j P 4n 8. By the second part of Lemma 3.1, we have n3(G) = jXj P 4n 8 P 3n 4. This contradicts our hypothesis, and so G is maximally 3-restricted edge connected. h 4. Reliability analysis Let G be an n-cube with n P 4. Then it is an n-regular connected vertex-transitive graph with order 2n, size e = n 2n1 and girth 4. From Lemmas 2.1 and 2.2, and Theorem 3.2, we deduce that k(G) = n, k2(G) = 2n 2 and k3(G) = 3n 4. Let S be
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an edge cut of size h. Then e P h P n. If 2n 3 P h P n then G S consists of two components, one of which is an isolated vertex; if 3n 5 P h P 2n 2 then either G S consists of three components, two of which are isolated vertices, or G S consists of two components, one of which is an isolated vertex or isolated edge. Let E(u) denote the set of edge cuts of size at most 3n 5 of graph G that separates vertex u from G, and E(f) be the set of edge cuts of size at most 3n 5 of graph G that separates edge f from G. With previous discussion and symbols, we have: Lemma 4.1. Let u be a vertex of n-cube G and f be one of its edges. If n > 4 then n If define ¼ 0 when m < 0, we have the following result. m
S
u2VðGÞ EðuÞ
\
S
f 2EðGÞ Eðf Þ
¼ ;.
Theorem 4.2. Let G be an n-cube with size e and Ch be the number of its edge cuts of size h. If n > 4 then
8 0; if h < n > > > > e n > > > 2n ; if 2n 3 P h P n > > hn > > > > > > < 2n e n þ e e 2n þ 1 hn h 2n þ 2 Ch ¼ > n > > e 2n þ 1 2 > > > e 2 > > h 2n þ 2 2 > > > > > e 2n > > : ; if 3n 5 P h P 2n 2 h 2n Proof. From Lemma 2.1 it follows that Ch = 0 when h < n. Let S be an edge cut of size h. If 2n 3 > h > n then G S consists of two components, one of which is an isolated vertex. Let Ch(u) denote the number of edge cuts of size h that separate vertex u from G. Then
C h ðuÞ ¼
en hn
It follows that
C h ¼ 2n
:
en hn
:
If 3n 5 > h > 2n 2, then the set of edge cuts of size h consists of two disjoint subsets by Lemma 4.1 and its proof. The first one formed by those edge cuts that separate one or two vertices from G. This set has cardinality
[ X X EðuÞ ¼ C ðuÞ jEðuÞ \ Eðv Þj: u2VðGÞ h u2VðGÞ u;v 2VðGÞ
If vertex u and
v are adjacent to each other, then
jEðuÞ \ Eðv Þj ¼
e 2n þ 1 h 2n þ 1
:
If vertex u is not adjacent to v, then
jEðuÞ \ Eðv Þj ¼
e 2n h 2n
:
Therefore
[ en e 2n þ 1 2n e 2n EðuÞ ¼ 2n e e : u2VðGÞ hn h 2n þ 1 2 h 2n
ð2Þ
Each of the second kind edge cuts separates an isolated edge from G. Hence, the number of this kind edge cuts is
e
e 2n þ 1 h 2n þ 2
:
The second part of the theorem follows from the combination of (2) and (3).
ð3Þ h
5. Conclusion Let G be an n-cube with n > 4. If its nodes do not fail and edges fail independently with equal probability p < 1, then the coefficients Ch in its reliability polynomial can be calculated according to Theorem 4.2 when 3n5 > h > 2n 2, and Ch = 0 when h < n. Hence, an approximation of precision O(p3n-4) of R(G, p) is derived when p ? 0.
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Acknowledgement Supported by National Natural Science Foundation of China (Grant No. 10801091); NSF of Guandong Province (Grant Nos. 8152902001000004; 9151051501000072). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
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