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The generalized 3-connectivity of star graphs and bubble-sort graphs
Applied Mathematics and Computation 274 (2016) 41–46
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
The generalized 3-connectivity of star graphs and bubble-sort graphs Shasha Li a, Jianhua Tu b,∗, Chenyan Yu a a b
Department of Fundamental Course, Ningbo Institute of Technology, Zhejiang University, Ningbo 315100, China Department of Mathematics, Beijing University of Chemical Technology, Beijing 100029, China
a r t i c l e
i n f o
MSC: 05C40 90C35 Keywords: Generalized 3-connectivity Internally disjoint trees Cayley graphs Star graphs Bubble-sort graphs
a b s t r a c t For S ⊆ G, let κ (S) denote the maximum number r of edge-disjoint trees T1 , T2 , . . . , Tr in G such that V (Ti ) ∩ V (T j ) = S for any i, j ∈ {1, 2, . . . , r} and i = j. For every 2 ≤ k ≤ n, the generalized k-connectivity of G κ k (G) is defined as the minimum κ (S) over all k-subsets S of vertices, i.e., κk (G) = min {κ(S)|S ⊆ V (G) and |S| = k}. Clearly, κ 2 (G) corresponds to the traditional connectivity of G. The generalized k-connectivity can serve for measuring the capability of a network G to connect any k vertices in G. Cayley graphs have been used extensively to design interconnection networks. In this paper, we restrict our attention to two classes of Cayley graphs, the star graphs Sn and the bubble-sort graphs Bn , and investigate the generalized 3-connectivity of Sn and Bn . We show that κ3 (Sn ) = n − 2 and κ3 (Bn ) = n − 2. © 2015 Elsevier Inc. All rights reserved.
1. Introduction The traditional connectivity κ (G) of a graph G is defined as the minimum cardinality of a subset Q of vertices of G such that G − Q is disconnected or trivial. A graph G is said to be k-connected if κ (G) ≥ k. Two distinct paths are internally disjoint if they have no internal vertices in common. A well-known theorem of Whitney [20] provides an equivalent definition of connectivity. For each 2-subset S = {u, v} of vertices of G, let κ (S) denote the maximum number of internally disjoint (u, v)-paths in G. Then κ(G) = min {κ(S)|S ⊆ V and |S| = 2}. As a means of strengthening the connectivity, the generalized connectivity was introduced, among the same definition given by other authors, by Hager [3,4]. Let G be a nontrivial connected graph of order n. For S ⊆ V(G), T1 and T2 are two internally disjoint trees connecting S if T1 and T2 are edge-disjoint and V (T1 ) ∩ V (T2 ) = S (note that the two trees are vertex-disjoint in G − S). Let κ (S) denote the maximum number of internally disjoint trees connecting S in G. The generalizedk-connectivity of G, denoted by κ k (G), is then defined by κk (G) = min {κ(S)|S ⊆ V (G) and |S| = k}, where 2 ≤ k ≤ n. Thus, when k = 2, the generalized 2-connectivity κ 2 (G) of G is exactly the connectivity κ (G), namely κ2 (G) = κ(G). There have been many results on the generalized connectivity, see [8–14] and a survey [15]. The concept of generalized connectivity is related to another generalization of traditional connectivity, called rainbow connection number. Let G be a nontrivial connected graph on which an edge-coloring c : E (G) → {1, 2, . . . , n}, is defined, where adjacent edges may be colored the same. A path is rainbow if no two edges of it are colored the same. An edgecoloring graph G is rainbow connected if any two vertices are connected by a rainbow path. We define the rainbow connection number of a connected graph G, denoted by rc(G), as the smallest number of colors that are needed in order to make G rainbow connected. The rainbow connection number has been widely studied [6,16–18]. ∗
Corresponding author. Tel.: +86 18500975081; fax: +86 01064430220. E-mail addresses: [email protected] (S. Li), [email protected] (J. Tu), [email protected] (C. Yu).
http://dx.doi.org/10.1016/j.amc.2015.11.016 0096-3003/© 2015 Elsevier Inc. All rights reserved.
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S. Li et al. / Applied Mathematics and Computation 274 (2016) 41–46
The underlying topology of a computer interconnection network can be modeled by a graph G, and the connectivity κ (G) of G is an important measure for fault tolerance of the network. In general, the larger κ (G) is, the more reliable the network is. However, if one wants to know how tough a network can be, for the connection of a set of vertices, then the generalized k-connectivity can serve for measuring the capability of a network G to connect any k vertices in G. Since Cayley graphs have been used extensively to design interconnection networks, the study of the generalized kconnectivity of Cayley graphs is very significative. Let X be a group and S be a subset of X. The Cayley digraph Cay(X, S) is a digraph with vertex set X and arc set {(g, gs)|g ∈ X, s ∈ S}. Clearly, if S = S−1 , where S−1 = {s−1 |s ∈ S}, then Cay(X, S) can be made into an undirected graph. Cayley (di)graphs have a lot of properties which are desirable in an interconnection network [5,7]: vertex symmetry makes it possible to use the same routing protocols and communication schemes at all nodes; hierarchical structure facilitates recursive constructions; high fault tolerance implies robustness, among others. Now, we consider Cayley graphs Cay(X, S) when the group X is a permutation group. Denote by Sym(n) the group of all permutations on {1, . . . , n}. Let ( p1 p2 . . . pn ) denote a permutation on {1, . . . , n} and (ij), which is called a transposition, denote the permutation that swaps the objects at positions i and j (not swapping element i and j), that is, ( p1 . . . pi . . . p j . . . pn )(i j) = ( p1 . . . p j . . . pi . . . pn ). Let T be a set of transpositions and G(T ) be the graph on n vertices {1, 2, . . . , n} such that there is an edge ij in G(T ) if and only if the transposition (i j) ∈ T . The graph G(T ) is called the transposition generating graph of Cay(Sym(n), T ). Moreover, if G(T ) is a tree, we call G(T ) a transposition tree and denote Cay(Sym(n), T ) by n . Specially, if G(T ) ∼ = K1,n−1 , then Cay(Sym(n), T ) is called a star graph Sn ; and Cay(Sym(n), T ) is called a bubble-sort graph Bn if G(T ) ∼ = Pn . In this paper, we study the generalized 3-connectivity of the star graph Sn and the bubble-sort graph Bn , and show that κ3 (Sn ) = n − 2 and κ3 (Bn ) = n − 2. 2. Preliminaries We first introduce some notation and results that will be used throughout the paper. We consider finite and simple graphs G. V(G) and E(G) denote its vertex set and its edge set respectively. For v ∈ V (G), denote by NG (v) the set of neighbors of v in G. For a subset U ⊆ V(G), let N(U) := ( ∪ u∈U N(u))ࢨU, and the subgraph induced by U is denoted by G[U]. Sometimes, we use a graph itself to represent its vertex set, for instance, N(G1 ) means N(V(G1 )), where G1 is a subgraph of G. Lemma 2.1 ([14]). Let G be a connected graph with minimum degree δ . Then κ 3 (G) ≤ δ . In particular, if there are two adjacent vertices of degree δ , then κ3 (G) ≤ δ − 1. Lemma 2.2 ([14]). Let G be a connected graph with n vertices. For every two integers k and r with k ≥ 0 and r ∈ {0, 1, 2, 3}, if
κ(G) = 4k + r, then κ3 (G) ≥ 3k + 2r . Moreover, the lower bound is sharp.
Lemma 2.3 (The Fan Lemma [1], p. 214). Let G be a k-connected graph, x a vertex of G, and let Y ⊆ V − {x} be a set of at least k vertices of G. Then there exists a k-fan in G from x to Y, namely there exists a family of k internally disjoint (x, Y)-paths whose terminal vertices are distinct in Y. Recall that n = Cay(Sym(n), T ) represents the Cayley graphs generated by transposition trees G(T ). The Cayley graphs n are (n − 1)-regular bipartite graphs and have n! vertices, see [7] for the details. Without loss of generality, we assume that for the star graph Sn , the transposition generating graph is G(T ) = {{1, . . . , n}, {12, 13, . . . , 1n}} and for the bubble-sort graph Bn , the transposition generating graph is G(T ) = {{1, . . . , n}, {12, 23, . . . , (n − 1)n}} throughout the paper. Now we give some useful properties, which can be found in [2,19,21]. Lemma 2.4 ([2,21]). κ(n ) = n − 1. Thus, κ(Sn ) = n − 1 and κ(Bn ) = n − 1. Property 2.1. [19] For n , if n is a leaf of G(T ), then n can be decomposed into n disjoint copies of n−1 , say 1 , 2 , . . . , n , where i n−1 is an induced subgraph by vertex set {( p1 p2 . . . pn−1 i)|( p1 . . . pn−1 ) ranges over all permun−1 n−1 n−1 1 2 n . ⊕ n−1 ⊕ . . . ⊕ n−1 tations of {1, . . . , n}\{i}}. We denote this decomposition by n = n−1 1 2 n Thus, by Property 2.1 Sn = Sn−1 ⊕ Sn−1 ⊕ · · · ⊕ Sn−1 and Bn = B1n−1 ⊕ B2n−1 ⊕ · · · ⊕ Bnn−1 i Property 2.2. [2] Consider the Gayley graphs n . Let (tn) ∈ T be a pendant edge of G(T ). For any vertex u of n−1 , u(tn) is the i i , is called the out-neighbor of u, written u . We call the neighbors of u in n−1 the in-neighbors unique neighbor of u outside of n−1
i have different out-neighbors. Hence, there are exactly (n − 2)! independent edges between of u. Any two distinct vertices of n−1
j j i i n−1 and n−1 if i = j, that is, |N(n−1 ) ∩ V (n−1 )| = (n − 2)! if i = j.
We give the following result.
S. Li et al. / Applied Mathematics and Computation 274 (2016) 41–46
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1 2 n Lemma 2.5. Let (tn) ∈ T be a pendant edge of G(T ) and n = n−1 ⊕ n−1 ⊕ . . . ⊕ n−1 be the decomposition of n . If n ≥ 3, then j i κ(n−1 ⊕ n−1 ) = n − 2,
i i for any two distinct integers i, j ∈ {1, . . . , n}, where n−1 ⊕ n−1 is the induced subgraph in n by V (n−1 ) ∪ V (n−1 ). j
j
j i ⊕ n−1 . Since n ≥ 3, there must exist a vertex v in ¯ such that the out-neighbor of v is outside of ¯ . Thus, Proof. Let ¯ := n−1 d¯ (v) = n − 2 and it follows that κ(¯ ) ≤ δ(¯ ) = n − 2. ¯ ) ≥ 1. Now, we show that κ(¯ ) ≥ n − 2. For n = 3, by Lemma 2.4 and Property 2.2, ¯ is obviously connected, and hence κ( For n ≥ 4, let v1 and v2 be any two vertices of ¯ . We distinguish the following two cases to prove that there are at least n − 2 internally disjoint paths between v1 and v2 , i.e., κ({v1 , v2 }) ≥ n − 2, and hence κ(¯ ) ≥ n − 2.
Case 1: v1 , v2 belong to the same copy. i i i ∼ . By the fact that n−1 ) = κ(n−1 ) = n − 2, and Without loss of generality, let v1 , v2 ∈ n−1 = n−1 and Lemma 2.4, κ(n−1 hence κ({v1 , v2 }) ≥ n − 2. Case 2: v1 , v2 belong to different copies. j i i and v2 ∈ n−1 . Select n − 2 vertices u1 , u2 , . . . , un−2 in n−1 such that for every vertex uk (1 ≤ We assume that v1 ∈ n−1 k ≤ n − 2), element j is at position t, namely, uk = ( pk1 . . . pkt−1 jpkt+1 . . . i). This can be done because (n − 2)! ≥ (n − 2). Let S :=
j {u1 , u2 , . . . , un−2 }. Clearly, the all out-neighbors of vertices in S are in n−1 . Let S = {u1 , u2 , . . . , un−2 } be the out-neighbors of ¯ vertices in S. Thus, uk uk ∈ E ( ), for every k ∈ {1, . . . , n − 2}. i ) = n − 2. By Lemma 2.3, there exists a family of n − 2 internally disjoint (v1 , S)-paths P1 , P2 , . . . , Pn−2 such Recall that κ(n−1 that the terminal vertex of Pi is ui for every i ∈ {1, 2, . . . , n − 2}. Note that it is possible that v1 ∈ S. If so, there exists a (v1 , S)-path that contains only one vertex v1 . such that the terminal vertex of Pi Similarly, there exists a family of n − 2 internally disjoint (v2 , S )-paths P1 , P2 , . . . , Pn−2 is ui for every i ∈ {1, 2, . . . , n − 2}. Now, we can obtain n − 2 internally disjoint (v1 , v2 )-paths: v1 P1 u1 u1 P1 v2 , v1 P2 u2 u2 P2 v2 , . . . , v , and hence κ({v , v }) ≥ n − 2. v1 Pn−2 un−2 un−2 Pn−2 2 1 2 In conclusion, κ(¯ ) = n − 2. The proof is complete.
By the proof of Lemma 2.5, in fact we can also obtain the following lemma. 1 2 n Lemma 2.6. Let (tn) ∈ T be a pendant edge of G(T ) and n = n−1 ⊕ n−1 ⊕ . . . ⊕ n−1 be the decomposition of n . For every
i i ∈ {1, 2, . . . , n}, let i := n [V (n ) \ V (n−1 )]. If n ≥ 3, then for every i ∈ {1, 2, . . . , n},
κ(i ) = n − 2. 3. κ3 (Sn ) In this section, we determine κ 3 (Sn ). First, we give the following claim and it is not hard to see that the claim indeed holds. 1 2 n Claim 3.1. Let Sn be a star graph and Sn−1 ⊕ Sn−1 ⊕ · · · ⊕ Sn−1 be the decomposition of Sn . Consider any vertex v ∈ V (Sn ). Suppose i i v ∈ Sn−1 . Let Nin (v) := NSi (v) ∪ {v}, i.e., Nin (v) is the closed neighborhood of v in Sn−1 . Then the n − 1 out-neighbors of vertices n−1
in Nin (v) are in different copies. In other words, for every j ∈ {1, . . . , n} and j = i, there is exactly one vertex in Nin (v) such that j the out-neighbor of the vertex is in Sn−1 . n . Then, To illustrate Claim 3.1 let us consider an example: v = (123 . . . n) ∈ Sn−1
Nin (v) =
{v(12), v(13), . . . , v(1(n − 1))} ∪ {v} = {(213 . . . n), (321 . . . n), . . . , ((n − 1)23 . . . 1n)} ∪ {(123 . . . n)}
the n − 1 out-neighbors of vertices in Nin (v) are
(n13 . . . 2), (n21 . . . 3), . . . , (n23 . . . 1(n − 1)), (n23 . . . 1). 2 , . . . , Sn−1 and S1 , respectively. These out-neighbors are contained in Sn−1 n−1 n−1 Now, we give the first main result.
Theorem 3.2. κ3 (Sn ) = n − 2, for any n ≥ 3. Proof. Since Sn is an (n − 1)-regular graph, by Lemma 2.1, κ3 (Sn ) ≤ δ − 1 = n − 2. Thus we just need to prove that κ3 (Sn ) ≥ n − 2. We prove by induction on n. For n = 3, obviously Sn is connected, and hence κ3 (Sn ) ≥ 1 = n − 2. For n = 4, by Lemma 2.4, κ(S4 ) = 4 − 1 = 3. By Lemma 2.2, κ3 (S4 ) ≥ 32 = 2 = n − 2. 1 2 n ⊕ Sn−1 ⊕ . . . , ⊕Sn−1 be the decomposition of Sn . Let v1 , v2 and v3 be any three vertices of Sn , Now suppose that n ≥ 5. Let Sn−1 and H := {v1 , v2 , v3 }. We distinguish three cases:
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S. Li et al. / Applied Mathematics and Computation 274 (2016) 41–46 i Case 1: v1 , v2 and v3 belong to the same copy Sn−1 . i ∼ Sn−1 . By the inductive hypothesis, κ3 (Si Note that S = n−1
n−1
) ≥ n − 3. That is to say, there are at least (n − 3) internally disjoint
i . trees connecting H in Sn−1
i )] is connected, and hence Let v1 , v2 and v3 be the out-neighbors of v1 , v2 and v3 , respectively. By Lemma 2.6, Sn [V (Sn ) \ V (Sn−1 contains a tree T connecting {v1 , v2 , v3 }. The tree T by adding three pendant edges v1 v1 , v2 v2 , v3 v3 to T is a tree connecting H i and V (T ) ∩ V (Sn−1 ) = H. Now, in this case there are at least n − 2 internally disjoint trees connecting H in Sn , and hence κ(H ) ≥ n − 2. Case 2: v1 , v2 and v3 belong to two copies. 1 2 . By Lemma 2.4, κ(S1 ) = n − 2, and hence there are n − 2 Without loss of generality, suppose that v1 , v2 ∈ Sn−1 and v3 ∈ Sn−1 n−1 1 internally disjoint (v1 , v2 )-paths P1 , P2 , . . . , Pn−2 in Sn−1 . Choose (n − 2) distinct vertices x1 , x2 , . . . , xn−2 from P1 , P2 , . . . , Pn−2 such that xi ∈ V(Pi ), for 1 ≤ i ≤ n − 2. Note that at most one of these paths, say P1 , has length 1, if so, we can choose x1 = v1 . Let xi be 1 have different out-neighbors. So the out-neighbor of xi , for all i ∈ {1, . . . , n − 2}. By Property 2.2, any two distinct vertices of Sn−1 X = {x1 , x2 , . . . , xn−2 } is a set of size (n − 2). 1 )] whose terBy Lemmas 2.6 and 2.3, there exist n − 2 internally disjoint (v3 , X )-paths P1 , P2 , . . . , Pn−2 in Sn [V (Sn ) \ V (Sn−1 minal vertices are distinct in X . Note that if v3 ∈ X , then there is a (v3 , X )-path that contains exactly one vertex v3 . Now, are n − 2 internally disjoint trees connecting H, and hence κ(H ) ≥ n − 2. T1 = P1 ∪ x1 x1 ∪ P1 , . . . , Tn−2 = Pn−2 ∪ xn−2 xn−2 ∪ Pn−2 Case 3: v1 , v2 and v3 belong to different copies, respectively. 1 , v ∈ S2 3 . Let N := N and v3 ∈ Sn−1 (vi ) ∪ {vi } for every i ∈ {1, 2, 3}. Without loss of generality, suppose that v1 ∈ Sn−1 2 i n−1 Si n−1
By Claim 3.1, for every i ∈ {1, 2, 3} and j ∈ {4, . . . , n}, there exists exactly one vertex in Ni , say ui , such that the out-neighbor j
is in Since is connected, we can obtain a tree connecting H for every j ∈ {4, . . . , n}. Now, of we can obtain n − 3 internally disjoint trees connecting H. i i := Sn−1 − ({u4i , . . . , uni } \ {vi }) for every i ∈ {1, 2, 3}. Since for every i ∈ {1, 2, 3} at most n − 3 vertices is deleted from Let Sn−1 i , by Lemma 2.4 κ(Si ) = n − 2 and Si is connected. S j ui
j Sn−1 .
n−1
j j j j ∪ {u1 , u2 , u3 } ∪ V (Sn−1 )
j Sn−1
n−1
n−1
k l and Sn−1 is at least (n − 2)! − 2(n − 3) ≥ On the other hand, for every k, l ∈ {1, 2, 3} and k = l, the number of edges between Sn−1 3 i 1 (by Property 2.2 and n ≥ 5). Thus, the induced subgraph in Sn by ∪i=1V (Sn−1 ) is connected and its spanning tree is a tree connecting H. Now, in this case there are at least (n − 2) internally disjoint trees connecting H. We conclude that κ(H ) ≥ n − 2, and hence κ3 (Sn ) = n − 2. The proof is complete.
4. κ3 (Bn ) In this section, we consider the bubble-sort graphs Bn . Obviously, for any vertex v ∈ V (Bn ), the unique out-neighbor of v is v((n − 1)n). Unfortunately, Bn does not satisfy Claim 3.1. For example, consider v = (123 . . . n) ∈ Bnn−1 . The (n − 2) in-neighbors of v are v(12) = (213 . . . (n − 1)n), v(23) = (132 . . . (n − 1)n), . . . , and v((n − 2), (n − 1)) = (123 . . . (n − 1), (n − 2)n), respectively. n−1 . Obviously, the all out-neighbors of v(12), v(23), . . . , v((n − 3)(n − 2)) are in Sn−1 In order to deal with the difference, we will replace the in-neighbors by internally disjoint paths. See details in the proof of the following theorem. Theorem 4.1. κ3 (Bn ) = n − 2, for any n ≥ 3. Proof. Since Bn is (n − 1)-regular, κ3 (Bn ) ≤ δ − 1 = n − 2. To complete the proof, it suffices to show that κ3 (Bn ) ≥ n − 2. We prove by induction on n. For n = 3, clearly κ3 (B3 ) ≥ 1 = n − 2. κ(B ) For n = 4, by Lemmas 2.2 and 2.4, κ3 (B4 ) ≥ 2 4 = 32 = 2. 1 2 Now suppose that n ≥ 5. Let Bn−1 ⊕ Bn−1 ⊕ . . . , ⊕Bnn−1 be the decomposition of Bn . Let v1 , v2 and v3 be any three vertices of Bn , and H := {v1 , v2 , v3 }. We distinguish three cases: Case 1: v1 , v2 and v3 belong to the same copy; Case 2: v1 , v2 and v3 belong to two copies; Case 3: v1 , v2 and v3 belong to different copies, respectively. The proofs of Cases 1 and 2 are the same as those of Cases 1 and 2 in Theorem 3.2. Thus we only give the proof of Case 3. Without loss of generality, suppose that v1 ∈ B1n−1 , v2 ∈ B2n−1 and v3 ∈ B3n−1 . Consider the vertex v1 = (i1 i2 . . . , in−1 1). Then there are the following n − 1 paths starting at the vertex v1 in B1n−1 .
P21 =: (i1 i2 i3 . . . , in−1 1)(i2 i1 i3 . . . , in−1 1)(i2 i3 i1 . . . , in−1 1) . . . (i2 i3 . . . , in−1 i1 1); P31 =: (i1 i2 i3 . . . , in−1 1)(i1 i3 i2 . . . , in−1 1) . . . (i1 i3 . . . , in−1 i2 1); ... 1 =: (i1 i2 i3 . . . , in−2 in−1 1)(i1 i2 i3 . . . , in−1 in−2 1) Pn−1
S. Li et al. / Applied Mathematics and Computation 274 (2016) 41–46
Pn1 := (i1 i2 i3 . . . , in−1 1)
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(only one vertex v1 ).
Fact 1: For every k, l ∈ {2, . . . , n} and k = l, V (Pk1 ) ∩ V (Pl1 ) = {v1 }; We prove the fact. W.l.o.g., suppose that k < l. For every vertex y ∈ V (Pk1 ) \ {v1 }, the k − 1 elements at positions 1, 2, . . . , k − 1 of y are i1 , i2 , . . . , ik−2 , ik . However, for every vertex z ∈ V (Pl1 ) \ {v1 }, the k − 1 elements at positions 1, 2, . . . , k − 1 of z are i1 , i2 , . . . , ik−2 , ik−1 . Thus, the fact indeed holds. Fact 2: Let
X 1 := {u1i |u1i is the terminal vertex of the path Pi1 for i ∈ {2, . . . , n}}. Then the out-neighbors of vertices in X1 are in B2n−1 , B3n−1 , . . . , Bnn−1 , respectively.
The second fact clearly holds. Furthermore, we can assume that the out-neighbor (u1i ) of u1i is in Bin−1 for every i ∈ {2, . . . , n}, otherwise we have to reorder these paths accordingly. Similarly, there are n − 1 paths P12 , P32 , . . . , Pn2 starting at the vertex v2 in B2n−1 and X 2 = {u21 , u23 , . . . , u2n } such that u2i is the
terminal vertex of Pi2 , and the out-neighbor (u2i ) of u2i is in Bin−1 for every i ∈ {1, 3, 4, . . . , n}. there are n − 1 paths P13 , P23 , P43 . . . , Pn3 starting at the vertex v3 in B3n−1 and X 3 = {u31 , u32 , u34 , . . . , u3n } such that u3i is the terminal
vertex of Pi3 , and the out-neighbor (u3i ) of u3i is in Bin−1 for every i ∈ {1, 2, 4, . . . , n}. Specially, the out-neighbor (u31 ) of u31 is in B1n−1 and the out-neighbor (u32 ) of u32 is in B2n−1 . 1 in B1 . Let t be the first vertex of the path P which is in U Since B1n−1 is connected, there is a ((u31 ) , v1 )-path P 1 1 1 k∈{2,...,n}V (Pk ). n−1 3 2 2 Likewise, there is a ((u ) , v )-path P in B . Let t be the first vertex of the path P which is in U V (P ). 2
2
2
n−1
2
Now, we distinguish two subcases. Subcase 3.1: t1 ∈ ∪k∈{2,3}V (Pk1 ) and t2 ∈ ∪k∈{1,3}V (Pk2 ).
2
k∈{1,3,...,n}
k
[(u3 ) , t ]) ∪ V (P 1 ) ∪ V (P 1 ) contains a (v , v )-path, where In this subcase, the induced subgraph in Sn by V (P13 ) ∪ V (P 1 1 3 1 2 3 1 3 3 P1 [(u1 ) , t1 ] is the subpath of the path P1 starting at (u1 ) and ending at t1 . [(u3 ) , t ] [(u3 ) , t ]) ∪ V (P 2 ) ∪ V (P 2 ) contains a (v , v )-path, where P Likewise, the induced subgraph in Sn by V (P23 ) ∪ V (P 2 2 3 2 2 2 1 3 2 2 3 is the subpath of the path P2 starting at (u2 ) and ending at t2 . The union of the (v3 , v1 )-path and (v3 , v2 )-path forms a tree connecting H. j On the other hand, for every j ∈ {4, . . . , n}, there exists a tree connecting V (Pj1 ) ∪ V (Pj2 ) ∪ V (Pj3 ) ∪ V (Bn−1 ).
Hence in this subcase we conclude that there are n − 2 internally disjoint trees connecting H in Bn and κ(H ) ≥ n − 2. Subcase 3.2: t1 ∈ ∪k∈{4,5,...,n}V (Pk1 ) or t2 ∈ ∪k∈{4,5,...,n}V (Pk2 ). W.l.o.g., suppose that t1 ∈ V (P41 ). Recall that v1 = (i1 i2 . . . , in−1 1). Clearly, at least one of in−1 = 2 and in−1 = 3 must hold. W.l.o.g., we assume that in−1 = 2. Consider the path P21 . Note that u12 is the terminally vertex of P21 and we can assume that u12 = (i1 i2 . . . , in−2 21). Suppose that il = 4. We now extend the path P21 starting from u12 as follows.
(i1 . . . , 4il+1 . . . in−2 21)(i1 . . . , il+1 4 . . . , in−2 21) . . . (i1 , . . . , il+1 . . . , in−2 241) 1 the extended path starting at v and ending Let u12 := (i1 , . . . , il+1 . . . , in−2 241). The out-neighbor of u12 is in B4n−1 . Denote by P 1 2 1 at u2 . We show that the following claim holds. 1 ) ∩ V (P 1 ) = {v }, for any j ∈ {3, 4, . . . , n}. Claim 4.2. V (P 1 2 j 1 ) ∩ V (P 1 )| ≥ 2. Proof by contradiction. Suppose that there exists k ∈ {3, 4, . . . , n} such that |V (P 2 k 1 1 1 1 1 ) \ V (P 1 ). If w = / V (P21 ). Thus, w ∈ V (P u12 , We assume that w ∈ V (P2 ) ∩ V (Pk ) and w = v1 . Since V (P2 ) ∩ V (Pk ) = {v1 }, w ∈ 2 2 1 then the element at position (n − 1) of w is 2. However, the element at position (n − 1) of each vertex in V (Pk ) is in−1 or k, a contradiction. If w = u12 = (i1 , . . . , il+1 . . . , in−2 241), then k = 4 and w = u14 . However, the element at position n − 2 of u14 is in−1 , a contradiction. The proof of the claim is complete. Similarly, if t2 ∈ V (Pl2 ) and l ∈ {4, 5, . . . , n}, we can extend the path P12 or the path P32 to obtain an extended path such that
the out-neighbor of the terminal vertex of the extended path is in Bln−1 and there is only one common vertex v2 between the extended path and the other paths. [(u3 ) , t ]) ∪ V (P 1 ) contains a (v , v )-path. Similarly, we can obtain a Now, the induced subgraph in Sn by V (P13 ) ∪ V (P 1 1 3 1 4 1 (v3 , v2 )-path. The union of the (v3 , v1 )-path and (v3 , v2 )-path forms a tree connecting H. At the same time, for every j ∈ j {4, 5, . . . , n}, there exists a tree connecting H ∪ V (Bn−1 ). The most important thing is that we can guarantee that these n − 2 trees connecting H are internally disjoint from the previous discussions. In conclusion, κ(H ) ≥ n − 2, and hence κ3 (Bn ) = n − 2. The proof is complete.
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5. Conclusion The generalized k-connectivity is a natural generalization of the traditional connectivity and can serve for measuring the capability of a network G to connect any k vertices in G. In this paper, we restrict our attention to two classes of Cayley graphs, the star graphs Sn and the bubble-sort graphs Bn . We investigate the generalized 3-connectivity of star graphs and bubblesort graphs and show that κ3 (Sn ) = n − 2 and κ3 (Bn ) = n − 2. For future work, it would be interesting to study the generalized connectivity of some other classes of Cayley graphs and some other network graphs. Acknowledgments The first author’s work was supported by t he National Science Foundation of China (no. 11301480) and the Natural Science Foundation of Ningbo, China (nos. 2014A610030, 2013A610067). This research work was done while the second author was visiting at the University of Texas at Dallas, supported by the National Natural Science Foundation of China (no. 11201021), Beijing Higher Education Young Elite Teacher Project (no. YETP0517), and a scholarship from the China Scholarship Council. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
J.A. Bondy, U.S.R. Murty, Graph Theory, GTM 244, Springer, 2008. E. Cheng, L. Liptácuteák, N. Shawash, Orienting Cayley graphs generated by transposition trees, Comput. Math. Appl. 55 (2008) 2662–2672. M. Hager, Pendant tree-connectivity, J. Comb. Theory 38 (1985) 179–189. M. Hager, Path-connectivity in graphs, Discrete Math. 59 (1986) 53–59. M. Heydemann, Cayley graphs and interconnection networks, in: G. Hahn, G. Sabidussi (Eds.), Graph Symmetry, Kluwer Academic Publishers, Netherlands, 1997, pp. 167–224 X. Huang, X. Li, Y. Shi, J. Yue, Y. Zhao, Rainbow connections for outerplanar graphs with diameter 2 and 3, Appl. Math. Comput. 242 (2014) 277–280. S. Lakshmivarahan, J. Jwo, S.K. Dhall, Symmetry in interconnection networks based on Cayley graphs of permutation groups: a survey, Parallel Comput. 19 (1993) 361–407. S. Li, X. Li, Note on the hardness of generalized connectivity, J. Comb. Optim. 24 (2012) 389–396. S. Li, W. Li, X. Li, The generalized connectivity of complete bipartite graphs, Ars Comb. 104 (2012) 65–79. S. Li, W. Li, X. Li, The generalized connectivity of complete equipartition 3-partite graphs, Bull. Malays. Math. Sci. Soc. 37 (1) (2014a) 103–121. H. Li, X. Li, Y. Mao, Y. Sun, Note on the generalized connectivity, Ars Comb. 114 (2014b) 193–202. S. Li, X. Li, Y. Shi, The minimal size of a graph with generalized connectivity κ3 = 2, Australas. J. Comb. 51 (2011) 209–220. H. Li, X. Li, Y. Sun, The generalized 3-connectivity of Cartesian product graphs, Discrete Math. Theor. Comput. Sci. 14 (1) (2012) 43–54. S. Li, X. Li, W. Zhou, Sharp bounds for the generalized connectivity κ 3 (g), Discrete Math. 310 (2010) 2147–2163. X. Li, Y. Mao, A survey on the generalized connectivity of graphs, arXiv:1207.1838v9. X. Li, Y. Shi, Rainbow connection in 3-connected graphs, Graphs Comb. 29 (5) (2013a) 1471–1475. X. Li, Y. Shi, On the rainbow vertex-connection, Discuss. Math. Graph Theory 33 (2) (2013b) 307–313. X. Li, Y. Shi, Y. Sun, Rainbow connections of graphs: A survey, Graphs Comb. 29 (1) (2013) 1–38. M. Wan, Z. Zhang, A kind of conditional vertex connectivity of star graphs, Appl. Math. Lett. 22 (2) (2009) 264–267. H. Whitney, Congruent graphs and the connectivity of graphs and the connectivity of graphs, Am. J. Math. 54 (1932) 150–168. W.H. Yang, H.Z. Li, J.X. Meng, Conditional connectivity of Cayley graphs generated by transposition trees, Inf. Process. Lett. 110 (2010) 1027–1030.
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
The generalized 3-connectivity of star graphs and bubble-sort graphs Shasha Li a, Jianhua Tu b,∗, Chenyan Yu a a b
Department of Fundamental Course, Ningbo Institute of Technology, Zhejiang University, Ningbo 315100, China Department of Mathematics, Beijing University of Chemical Technology, Beijing 100029, China
a r t i c l e
i n f o
MSC: 05C40 90C35 Keywords: Generalized 3-connectivity Internally disjoint trees Cayley graphs Star graphs Bubble-sort graphs
a b s t r a c t For S ⊆ G, let κ (S) denote the maximum number r of edge-disjoint trees T1 , T2 , . . . , Tr in G such that V (Ti ) ∩ V (T j ) = S for any i, j ∈ {1, 2, . . . , r} and i = j. For every 2 ≤ k ≤ n, the generalized k-connectivity of G κ k (G) is defined as the minimum κ (S) over all k-subsets S of vertices, i.e., κk (G) = min {κ(S)|S ⊆ V (G) and |S| = k}. Clearly, κ 2 (G) corresponds to the traditional connectivity of G. The generalized k-connectivity can serve for measuring the capability of a network G to connect any k vertices in G. Cayley graphs have been used extensively to design interconnection networks. In this paper, we restrict our attention to two classes of Cayley graphs, the star graphs Sn and the bubble-sort graphs Bn , and investigate the generalized 3-connectivity of Sn and Bn . We show that κ3 (Sn ) = n − 2 and κ3 (Bn ) = n − 2. © 2015 Elsevier Inc. All rights reserved.
1. Introduction The traditional connectivity κ (G) of a graph G is defined as the minimum cardinality of a subset Q of vertices of G such that G − Q is disconnected or trivial. A graph G is said to be k-connected if κ (G) ≥ k. Two distinct paths are internally disjoint if they have no internal vertices in common. A well-known theorem of Whitney [20] provides an equivalent definition of connectivity. For each 2-subset S = {u, v} of vertices of G, let κ (S) denote the maximum number of internally disjoint (u, v)-paths in G. Then κ(G) = min {κ(S)|S ⊆ V and |S| = 2}. As a means of strengthening the connectivity, the generalized connectivity was introduced, among the same definition given by other authors, by Hager [3,4]. Let G be a nontrivial connected graph of order n. For S ⊆ V(G), T1 and T2 are two internally disjoint trees connecting S if T1 and T2 are edge-disjoint and V (T1 ) ∩ V (T2 ) = S (note that the two trees are vertex-disjoint in G − S). Let κ (S) denote the maximum number of internally disjoint trees connecting S in G. The generalizedk-connectivity of G, denoted by κ k (G), is then defined by κk (G) = min {κ(S)|S ⊆ V (G) and |S| = k}, where 2 ≤ k ≤ n. Thus, when k = 2, the generalized 2-connectivity κ 2 (G) of G is exactly the connectivity κ (G), namely κ2 (G) = κ(G). There have been many results on the generalized connectivity, see [8–14] and a survey [15]. The concept of generalized connectivity is related to another generalization of traditional connectivity, called rainbow connection number. Let G be a nontrivial connected graph on which an edge-coloring c : E (G) → {1, 2, . . . , n}, is defined, where adjacent edges may be colored the same. A path is rainbow if no two edges of it are colored the same. An edgecoloring graph G is rainbow connected if any two vertices are connected by a rainbow path. We define the rainbow connection number of a connected graph G, denoted by rc(G), as the smallest number of colors that are needed in order to make G rainbow connected. The rainbow connection number has been widely studied [6,16–18]. ∗
Corresponding author. Tel.: +86 18500975081; fax: +86 01064430220. E-mail addresses: [email protected] (S. Li), [email protected] (J. Tu), [email protected] (C. Yu).
http://dx.doi.org/10.1016/j.amc.2015.11.016 0096-3003/© 2015 Elsevier Inc. All rights reserved.
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The underlying topology of a computer interconnection network can be modeled by a graph G, and the connectivity κ (G) of G is an important measure for fault tolerance of the network. In general, the larger κ (G) is, the more reliable the network is. However, if one wants to know how tough a network can be, for the connection of a set of vertices, then the generalized k-connectivity can serve for measuring the capability of a network G to connect any k vertices in G. Since Cayley graphs have been used extensively to design interconnection networks, the study of the generalized kconnectivity of Cayley graphs is very significative. Let X be a group and S be a subset of X. The Cayley digraph Cay(X, S) is a digraph with vertex set X and arc set {(g, gs)|g ∈ X, s ∈ S}. Clearly, if S = S−1 , where S−1 = {s−1 |s ∈ S}, then Cay(X, S) can be made into an undirected graph. Cayley (di)graphs have a lot of properties which are desirable in an interconnection network [5,7]: vertex symmetry makes it possible to use the same routing protocols and communication schemes at all nodes; hierarchical structure facilitates recursive constructions; high fault tolerance implies robustness, among others. Now, we consider Cayley graphs Cay(X, S) when the group X is a permutation group. Denote by Sym(n) the group of all permutations on {1, . . . , n}. Let ( p1 p2 . . . pn ) denote a permutation on {1, . . . , n} and (ij), which is called a transposition, denote the permutation that swaps the objects at positions i and j (not swapping element i and j), that is, ( p1 . . . pi . . . p j . . . pn )(i j) = ( p1 . . . p j . . . pi . . . pn ). Let T be a set of transpositions and G(T ) be the graph on n vertices {1, 2, . . . , n} such that there is an edge ij in G(T ) if and only if the transposition (i j) ∈ T . The graph G(T ) is called the transposition generating graph of Cay(Sym(n), T ). Moreover, if G(T ) is a tree, we call G(T ) a transposition tree and denote Cay(Sym(n), T ) by n . Specially, if G(T ) ∼ = K1,n−1 , then Cay(Sym(n), T ) is called a star graph Sn ; and Cay(Sym(n), T ) is called a bubble-sort graph Bn if G(T ) ∼ = Pn . In this paper, we study the generalized 3-connectivity of the star graph Sn and the bubble-sort graph Bn , and show that κ3 (Sn ) = n − 2 and κ3 (Bn ) = n − 2. 2. Preliminaries We first introduce some notation and results that will be used throughout the paper. We consider finite and simple graphs G. V(G) and E(G) denote its vertex set and its edge set respectively. For v ∈ V (G), denote by NG (v) the set of neighbors of v in G. For a subset U ⊆ V(G), let N(U) := ( ∪ u∈U N(u))ࢨU, and the subgraph induced by U is denoted by G[U]. Sometimes, we use a graph itself to represent its vertex set, for instance, N(G1 ) means N(V(G1 )), where G1 is a subgraph of G. Lemma 2.1 ([14]). Let G be a connected graph with minimum degree δ . Then κ 3 (G) ≤ δ . In particular, if there are two adjacent vertices of degree δ , then κ3 (G) ≤ δ − 1. Lemma 2.2 ([14]). Let G be a connected graph with n vertices. For every two integers k and r with k ≥ 0 and r ∈ {0, 1, 2, 3}, if
κ(G) = 4k + r, then κ3 (G) ≥ 3k + 2r . Moreover, the lower bound is sharp.
Lemma 2.3 (The Fan Lemma [1], p. 214). Let G be a k-connected graph, x a vertex of G, and let Y ⊆ V − {x} be a set of at least k vertices of G. Then there exists a k-fan in G from x to Y, namely there exists a family of k internally disjoint (x, Y)-paths whose terminal vertices are distinct in Y. Recall that n = Cay(Sym(n), T ) represents the Cayley graphs generated by transposition trees G(T ). The Cayley graphs n are (n − 1)-regular bipartite graphs and have n! vertices, see [7] for the details. Without loss of generality, we assume that for the star graph Sn , the transposition generating graph is G(T ) = {{1, . . . , n}, {12, 13, . . . , 1n}} and for the bubble-sort graph Bn , the transposition generating graph is G(T ) = {{1, . . . , n}, {12, 23, . . . , (n − 1)n}} throughout the paper. Now we give some useful properties, which can be found in [2,19,21]. Lemma 2.4 ([2,21]). κ(n ) = n − 1. Thus, κ(Sn ) = n − 1 and κ(Bn ) = n − 1. Property 2.1. [19] For n , if n is a leaf of G(T ), then n can be decomposed into n disjoint copies of n−1 , say 1 , 2 , . . . , n , where i n−1 is an induced subgraph by vertex set {( p1 p2 . . . pn−1 i)|( p1 . . . pn−1 ) ranges over all permun−1 n−1 n−1 1 2 n . ⊕ n−1 ⊕ . . . ⊕ n−1 tations of {1, . . . , n}\{i}}. We denote this decomposition by n = n−1 1 2 n Thus, by Property 2.1 Sn = Sn−1 ⊕ Sn−1 ⊕ · · · ⊕ Sn−1 and Bn = B1n−1 ⊕ B2n−1 ⊕ · · · ⊕ Bnn−1 i Property 2.2. [2] Consider the Gayley graphs n . Let (tn) ∈ T be a pendant edge of G(T ). For any vertex u of n−1 , u(tn) is the i i , is called the out-neighbor of u, written u . We call the neighbors of u in n−1 the in-neighbors unique neighbor of u outside of n−1
i have different out-neighbors. Hence, there are exactly (n − 2)! independent edges between of u. Any two distinct vertices of n−1
j j i i n−1 and n−1 if i = j, that is, |N(n−1 ) ∩ V (n−1 )| = (n − 2)! if i = j.
We give the following result.
S. Li et al. / Applied Mathematics and Computation 274 (2016) 41–46
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1 2 n Lemma 2.5. Let (tn) ∈ T be a pendant edge of G(T ) and n = n−1 ⊕ n−1 ⊕ . . . ⊕ n−1 be the decomposition of n . If n ≥ 3, then j i κ(n−1 ⊕ n−1 ) = n − 2,
i i for any two distinct integers i, j ∈ {1, . . . , n}, where n−1 ⊕ n−1 is the induced subgraph in n by V (n−1 ) ∪ V (n−1 ). j
j
j i ⊕ n−1 . Since n ≥ 3, there must exist a vertex v in ¯ such that the out-neighbor of v is outside of ¯ . Thus, Proof. Let ¯ := n−1 d¯ (v) = n − 2 and it follows that κ(¯ ) ≤ δ(¯ ) = n − 2. ¯ ) ≥ 1. Now, we show that κ(¯ ) ≥ n − 2. For n = 3, by Lemma 2.4 and Property 2.2, ¯ is obviously connected, and hence κ( For n ≥ 4, let v1 and v2 be any two vertices of ¯ . We distinguish the following two cases to prove that there are at least n − 2 internally disjoint paths between v1 and v2 , i.e., κ({v1 , v2 }) ≥ n − 2, and hence κ(¯ ) ≥ n − 2.
Case 1: v1 , v2 belong to the same copy. i i i ∼ . By the fact that n−1 ) = κ(n−1 ) = n − 2, and Without loss of generality, let v1 , v2 ∈ n−1 = n−1 and Lemma 2.4, κ(n−1 hence κ({v1 , v2 }) ≥ n − 2. Case 2: v1 , v2 belong to different copies. j i i and v2 ∈ n−1 . Select n − 2 vertices u1 , u2 , . . . , un−2 in n−1 such that for every vertex uk (1 ≤ We assume that v1 ∈ n−1 k ≤ n − 2), element j is at position t, namely, uk = ( pk1 . . . pkt−1 jpkt+1 . . . i). This can be done because (n − 2)! ≥ (n − 2). Let S :=
j {u1 , u2 , . . . , un−2 }. Clearly, the all out-neighbors of vertices in S are in n−1 . Let S = {u1 , u2 , . . . , un−2 } be the out-neighbors of ¯ vertices in S. Thus, uk uk ∈ E ( ), for every k ∈ {1, . . . , n − 2}. i ) = n − 2. By Lemma 2.3, there exists a family of n − 2 internally disjoint (v1 , S)-paths P1 , P2 , . . . , Pn−2 such Recall that κ(n−1 that the terminal vertex of Pi is ui for every i ∈ {1, 2, . . . , n − 2}. Note that it is possible that v1 ∈ S. If so, there exists a (v1 , S)-path that contains only one vertex v1 . such that the terminal vertex of Pi Similarly, there exists a family of n − 2 internally disjoint (v2 , S )-paths P1 , P2 , . . . , Pn−2 is ui for every i ∈ {1, 2, . . . , n − 2}. Now, we can obtain n − 2 internally disjoint (v1 , v2 )-paths: v1 P1 u1 u1 P1 v2 , v1 P2 u2 u2 P2 v2 , . . . , v , and hence κ({v , v }) ≥ n − 2. v1 Pn−2 un−2 un−2 Pn−2 2 1 2 In conclusion, κ(¯ ) = n − 2. The proof is complete.
By the proof of Lemma 2.5, in fact we can also obtain the following lemma. 1 2 n Lemma 2.6. Let (tn) ∈ T be a pendant edge of G(T ) and n = n−1 ⊕ n−1 ⊕ . . . ⊕ n−1 be the decomposition of n . For every
i i ∈ {1, 2, . . . , n}, let i := n [V (n ) \ V (n−1 )]. If n ≥ 3, then for every i ∈ {1, 2, . . . , n},
κ(i ) = n − 2. 3. κ3 (Sn ) In this section, we determine κ 3 (Sn ). First, we give the following claim and it is not hard to see that the claim indeed holds. 1 2 n Claim 3.1. Let Sn be a star graph and Sn−1 ⊕ Sn−1 ⊕ · · · ⊕ Sn−1 be the decomposition of Sn . Consider any vertex v ∈ V (Sn ). Suppose i i v ∈ Sn−1 . Let Nin (v) := NSi (v) ∪ {v}, i.e., Nin (v) is the closed neighborhood of v in Sn−1 . Then the n − 1 out-neighbors of vertices n−1
in Nin (v) are in different copies. In other words, for every j ∈ {1, . . . , n} and j = i, there is exactly one vertex in Nin (v) such that j the out-neighbor of the vertex is in Sn−1 . n . Then, To illustrate Claim 3.1 let us consider an example: v = (123 . . . n) ∈ Sn−1
Nin (v) =
{v(12), v(13), . . . , v(1(n − 1))} ∪ {v} = {(213 . . . n), (321 . . . n), . . . , ((n − 1)23 . . . 1n)} ∪ {(123 . . . n)}
the n − 1 out-neighbors of vertices in Nin (v) are
(n13 . . . 2), (n21 . . . 3), . . . , (n23 . . . 1(n − 1)), (n23 . . . 1). 2 , . . . , Sn−1 and S1 , respectively. These out-neighbors are contained in Sn−1 n−1 n−1 Now, we give the first main result.
Theorem 3.2. κ3 (Sn ) = n − 2, for any n ≥ 3. Proof. Since Sn is an (n − 1)-regular graph, by Lemma 2.1, κ3 (Sn ) ≤ δ − 1 = n − 2. Thus we just need to prove that κ3 (Sn ) ≥ n − 2. We prove by induction on n. For n = 3, obviously Sn is connected, and hence κ3 (Sn ) ≥ 1 = n − 2. For n = 4, by Lemma 2.4, κ(S4 ) = 4 − 1 = 3. By Lemma 2.2, κ3 (S4 ) ≥ 32 = 2 = n − 2. 1 2 n ⊕ Sn−1 ⊕ . . . , ⊕Sn−1 be the decomposition of Sn . Let v1 , v2 and v3 be any three vertices of Sn , Now suppose that n ≥ 5. Let Sn−1 and H := {v1 , v2 , v3 }. We distinguish three cases:
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S. Li et al. / Applied Mathematics and Computation 274 (2016) 41–46 i Case 1: v1 , v2 and v3 belong to the same copy Sn−1 . i ∼ Sn−1 . By the inductive hypothesis, κ3 (Si Note that S = n−1
n−1
) ≥ n − 3. That is to say, there are at least (n − 3) internally disjoint
i . trees connecting H in Sn−1
i )] is connected, and hence Let v1 , v2 and v3 be the out-neighbors of v1 , v2 and v3 , respectively. By Lemma 2.6, Sn [V (Sn ) \ V (Sn−1 contains a tree T connecting {v1 , v2 , v3 }. The tree T by adding three pendant edges v1 v1 , v2 v2 , v3 v3 to T is a tree connecting H i and V (T ) ∩ V (Sn−1 ) = H. Now, in this case there are at least n − 2 internally disjoint trees connecting H in Sn , and hence κ(H ) ≥ n − 2. Case 2: v1 , v2 and v3 belong to two copies. 1 2 . By Lemma 2.4, κ(S1 ) = n − 2, and hence there are n − 2 Without loss of generality, suppose that v1 , v2 ∈ Sn−1 and v3 ∈ Sn−1 n−1 1 internally disjoint (v1 , v2 )-paths P1 , P2 , . . . , Pn−2 in Sn−1 . Choose (n − 2) distinct vertices x1 , x2 , . . . , xn−2 from P1 , P2 , . . . , Pn−2 such that xi ∈ V(Pi ), for 1 ≤ i ≤ n − 2. Note that at most one of these paths, say P1 , has length 1, if so, we can choose x1 = v1 . Let xi be 1 have different out-neighbors. So the out-neighbor of xi , for all i ∈ {1, . . . , n − 2}. By Property 2.2, any two distinct vertices of Sn−1 X = {x1 , x2 , . . . , xn−2 } is a set of size (n − 2). 1 )] whose terBy Lemmas 2.6 and 2.3, there exist n − 2 internally disjoint (v3 , X )-paths P1 , P2 , . . . , Pn−2 in Sn [V (Sn ) \ V (Sn−1 minal vertices are distinct in X . Note that if v3 ∈ X , then there is a (v3 , X )-path that contains exactly one vertex v3 . Now, are n − 2 internally disjoint trees connecting H, and hence κ(H ) ≥ n − 2. T1 = P1 ∪ x1 x1 ∪ P1 , . . . , Tn−2 = Pn−2 ∪ xn−2 xn−2 ∪ Pn−2 Case 3: v1 , v2 and v3 belong to different copies, respectively. 1 , v ∈ S2 3 . Let N := N and v3 ∈ Sn−1 (vi ) ∪ {vi } for every i ∈ {1, 2, 3}. Without loss of generality, suppose that v1 ∈ Sn−1 2 i n−1 Si n−1
By Claim 3.1, for every i ∈ {1, 2, 3} and j ∈ {4, . . . , n}, there exists exactly one vertex in Ni , say ui , such that the out-neighbor j
is in Since is connected, we can obtain a tree connecting H for every j ∈ {4, . . . , n}. Now, of we can obtain n − 3 internally disjoint trees connecting H. i i := Sn−1 − ({u4i , . . . , uni } \ {vi }) for every i ∈ {1, 2, 3}. Since for every i ∈ {1, 2, 3} at most n − 3 vertices is deleted from Let Sn−1 i , by Lemma 2.4 κ(Si ) = n − 2 and Si is connected. S j ui
j Sn−1 .
n−1
j j j j ∪ {u1 , u2 , u3 } ∪ V (Sn−1 )
j Sn−1
n−1
n−1
k l and Sn−1 is at least (n − 2)! − 2(n − 3) ≥ On the other hand, for every k, l ∈ {1, 2, 3} and k = l, the number of edges between Sn−1 3 i 1 (by Property 2.2 and n ≥ 5). Thus, the induced subgraph in Sn by ∪i=1V (Sn−1 ) is connected and its spanning tree is a tree connecting H. Now, in this case there are at least (n − 2) internally disjoint trees connecting H. We conclude that κ(H ) ≥ n − 2, and hence κ3 (Sn ) = n − 2. The proof is complete.
4. κ3 (Bn ) In this section, we consider the bubble-sort graphs Bn . Obviously, for any vertex v ∈ V (Bn ), the unique out-neighbor of v is v((n − 1)n). Unfortunately, Bn does not satisfy Claim 3.1. For example, consider v = (123 . . . n) ∈ Bnn−1 . The (n − 2) in-neighbors of v are v(12) = (213 . . . (n − 1)n), v(23) = (132 . . . (n − 1)n), . . . , and v((n − 2), (n − 1)) = (123 . . . (n − 1), (n − 2)n), respectively. n−1 . Obviously, the all out-neighbors of v(12), v(23), . . . , v((n − 3)(n − 2)) are in Sn−1 In order to deal with the difference, we will replace the in-neighbors by internally disjoint paths. See details in the proof of the following theorem. Theorem 4.1. κ3 (Bn ) = n − 2, for any n ≥ 3. Proof. Since Bn is (n − 1)-regular, κ3 (Bn ) ≤ δ − 1 = n − 2. To complete the proof, it suffices to show that κ3 (Bn ) ≥ n − 2. We prove by induction on n. For n = 3, clearly κ3 (B3 ) ≥ 1 = n − 2. κ(B ) For n = 4, by Lemmas 2.2 and 2.4, κ3 (B4 ) ≥ 2 4 = 32 = 2. 1 2 Now suppose that n ≥ 5. Let Bn−1 ⊕ Bn−1 ⊕ . . . , ⊕Bnn−1 be the decomposition of Bn . Let v1 , v2 and v3 be any three vertices of Bn , and H := {v1 , v2 , v3 }. We distinguish three cases: Case 1: v1 , v2 and v3 belong to the same copy; Case 2: v1 , v2 and v3 belong to two copies; Case 3: v1 , v2 and v3 belong to different copies, respectively. The proofs of Cases 1 and 2 are the same as those of Cases 1 and 2 in Theorem 3.2. Thus we only give the proof of Case 3. Without loss of generality, suppose that v1 ∈ B1n−1 , v2 ∈ B2n−1 and v3 ∈ B3n−1 . Consider the vertex v1 = (i1 i2 . . . , in−1 1). Then there are the following n − 1 paths starting at the vertex v1 in B1n−1 .
P21 =: (i1 i2 i3 . . . , in−1 1)(i2 i1 i3 . . . , in−1 1)(i2 i3 i1 . . . , in−1 1) . . . (i2 i3 . . . , in−1 i1 1); P31 =: (i1 i2 i3 . . . , in−1 1)(i1 i3 i2 . . . , in−1 1) . . . (i1 i3 . . . , in−1 i2 1); ... 1 =: (i1 i2 i3 . . . , in−2 in−1 1)(i1 i2 i3 . . . , in−1 in−2 1) Pn−1
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Pn1 := (i1 i2 i3 . . . , in−1 1)
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(only one vertex v1 ).
Fact 1: For every k, l ∈ {2, . . . , n} and k = l, V (Pk1 ) ∩ V (Pl1 ) = {v1 }; We prove the fact. W.l.o.g., suppose that k < l. For every vertex y ∈ V (Pk1 ) \ {v1 }, the k − 1 elements at positions 1, 2, . . . , k − 1 of y are i1 , i2 , . . . , ik−2 , ik . However, for every vertex z ∈ V (Pl1 ) \ {v1 }, the k − 1 elements at positions 1, 2, . . . , k − 1 of z are i1 , i2 , . . . , ik−2 , ik−1 . Thus, the fact indeed holds. Fact 2: Let
X 1 := {u1i |u1i is the terminal vertex of the path Pi1 for i ∈ {2, . . . , n}}. Then the out-neighbors of vertices in X1 are in B2n−1 , B3n−1 , . . . , Bnn−1 , respectively.
The second fact clearly holds. Furthermore, we can assume that the out-neighbor (u1i ) of u1i is in Bin−1 for every i ∈ {2, . . . , n}, otherwise we have to reorder these paths accordingly. Similarly, there are n − 1 paths P12 , P32 , . . . , Pn2 starting at the vertex v2 in B2n−1 and X 2 = {u21 , u23 , . . . , u2n } such that u2i is the
terminal vertex of Pi2 , and the out-neighbor (u2i ) of u2i is in Bin−1 for every i ∈ {1, 3, 4, . . . , n}. there are n − 1 paths P13 , P23 , P43 . . . , Pn3 starting at the vertex v3 in B3n−1 and X 3 = {u31 , u32 , u34 , . . . , u3n } such that u3i is the terminal
vertex of Pi3 , and the out-neighbor (u3i ) of u3i is in Bin−1 for every i ∈ {1, 2, 4, . . . , n}. Specially, the out-neighbor (u31 ) of u31 is in B1n−1 and the out-neighbor (u32 ) of u32 is in B2n−1 . 1 in B1 . Let t be the first vertex of the path P which is in U Since B1n−1 is connected, there is a ((u31 ) , v1 )-path P 1 1 1 k∈{2,...,n}V (Pk ). n−1 3 2 2 Likewise, there is a ((u ) , v )-path P in B . Let t be the first vertex of the path P which is in U V (P ). 2
2
2
n−1
2
Now, we distinguish two subcases. Subcase 3.1: t1 ∈ ∪k∈{2,3}V (Pk1 ) and t2 ∈ ∪k∈{1,3}V (Pk2 ).
2
k∈{1,3,...,n}
k
[(u3 ) , t ]) ∪ V (P 1 ) ∪ V (P 1 ) contains a (v , v )-path, where In this subcase, the induced subgraph in Sn by V (P13 ) ∪ V (P 1 1 3 1 2 3 1 3 3 P1 [(u1 ) , t1 ] is the subpath of the path P1 starting at (u1 ) and ending at t1 . [(u3 ) , t ] [(u3 ) , t ]) ∪ V (P 2 ) ∪ V (P 2 ) contains a (v , v )-path, where P Likewise, the induced subgraph in Sn by V (P23 ) ∪ V (P 2 2 3 2 2 2 1 3 2 2 3 is the subpath of the path P2 starting at (u2 ) and ending at t2 . The union of the (v3 , v1 )-path and (v3 , v2 )-path forms a tree connecting H. j On the other hand, for every j ∈ {4, . . . , n}, there exists a tree connecting V (Pj1 ) ∪ V (Pj2 ) ∪ V (Pj3 ) ∪ V (Bn−1 ).
Hence in this subcase we conclude that there are n − 2 internally disjoint trees connecting H in Bn and κ(H ) ≥ n − 2. Subcase 3.2: t1 ∈ ∪k∈{4,5,...,n}V (Pk1 ) or t2 ∈ ∪k∈{4,5,...,n}V (Pk2 ). W.l.o.g., suppose that t1 ∈ V (P41 ). Recall that v1 = (i1 i2 . . . , in−1 1). Clearly, at least one of in−1 = 2 and in−1 = 3 must hold. W.l.o.g., we assume that in−1 = 2. Consider the path P21 . Note that u12 is the terminally vertex of P21 and we can assume that u12 = (i1 i2 . . . , in−2 21). Suppose that il = 4. We now extend the path P21 starting from u12 as follows.
(i1 . . . , 4il+1 . . . in−2 21)(i1 . . . , il+1 4 . . . , in−2 21) . . . (i1 , . . . , il+1 . . . , in−2 241) 1 the extended path starting at v and ending Let u12 := (i1 , . . . , il+1 . . . , in−2 241). The out-neighbor of u12 is in B4n−1 . Denote by P 1 2 1 at u2 . We show that the following claim holds. 1 ) ∩ V (P 1 ) = {v }, for any j ∈ {3, 4, . . . , n}. Claim 4.2. V (P 1 2 j 1 ) ∩ V (P 1 )| ≥ 2. Proof by contradiction. Suppose that there exists k ∈ {3, 4, . . . , n} such that |V (P 2 k 1 1 1 1 1 ) \ V (P 1 ). If w = / V (P21 ). Thus, w ∈ V (P u12 , We assume that w ∈ V (P2 ) ∩ V (Pk ) and w = v1 . Since V (P2 ) ∩ V (Pk ) = {v1 }, w ∈ 2 2 1 then the element at position (n − 1) of w is 2. However, the element at position (n − 1) of each vertex in V (Pk ) is in−1 or k, a contradiction. If w = u12 = (i1 , . . . , il+1 . . . , in−2 241), then k = 4 and w = u14 . However, the element at position n − 2 of u14 is in−1 , a contradiction. The proof of the claim is complete. Similarly, if t2 ∈ V (Pl2 ) and l ∈ {4, 5, . . . , n}, we can extend the path P12 or the path P32 to obtain an extended path such that
the out-neighbor of the terminal vertex of the extended path is in Bln−1 and there is only one common vertex v2 between the extended path and the other paths. [(u3 ) , t ]) ∪ V (P 1 ) contains a (v , v )-path. Similarly, we can obtain a Now, the induced subgraph in Sn by V (P13 ) ∪ V (P 1 1 3 1 4 1 (v3 , v2 )-path. The union of the (v3 , v1 )-path and (v3 , v2 )-path forms a tree connecting H. At the same time, for every j ∈ j {4, 5, . . . , n}, there exists a tree connecting H ∪ V (Bn−1 ). The most important thing is that we can guarantee that these n − 2 trees connecting H are internally disjoint from the previous discussions. In conclusion, κ(H ) ≥ n − 2, and hence κ3 (Bn ) = n − 2. The proof is complete.
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5. Conclusion The generalized k-connectivity is a natural generalization of the traditional connectivity and can serve for measuring the capability of a network G to connect any k vertices in G. In this paper, we restrict our attention to two classes of Cayley graphs, the star graphs Sn and the bubble-sort graphs Bn . We investigate the generalized 3-connectivity of star graphs and bubblesort graphs and show that κ3 (Sn ) = n − 2 and κ3 (Bn ) = n − 2. For future work, it would be interesting to study the generalized connectivity of some other classes of Cayley graphs and some other network graphs. Acknowledgments The first author’s work was supported by t he National Science Foundation of China (no. 11301480) and the Natural Science Foundation of Ningbo, China (nos. 2014A610030, 2013A610067). This research work was done while the second author was visiting at the University of Texas at Dallas, supported by the National Natural Science Foundation of China (no. 11201021), Beijing Higher Education Young Elite Teacher Project (no. YETP0517), and a scholarship from the China Scholarship Council. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
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