Strong Menger connectedness of augmented k -ary n -cubes
SStrong Menger connectedness of augmented k -ary n -cubes Mei-Mei Gu , ∗ Jou-Ming Chang † Rong-Xia Hao , ‡ Department of Mathematics, Beijing Jiaotong University, Beijing 100044, P.R. China Faculty of Mathematics and Physics, Charles University, Prague 11800, Czech Republic Institute of Information and Decision Sciences,National Taipei University of Business, Taipei 10051, Taiwan
Abstract
A connected graph G is called strongly Menger (edge) connected if for any two distinct vertices x, y of G , there are min { deg G ( x ) , deg G ( y ) } vertex(edge)-disjoint paths between x and y . In this paper,we consider strong Menger (edge) connectedness of the augmented k -ary n -cube AQ n,k , which is avariant of k -ary n -cube Q kn . By exploring the topological proprieties of AQ n,k , we show that AQ n, for n ≥ AQ n,k for n ≥ k ≥
4) is still strongly Menger connected even when there are4 n − n −
8) faulty vertices and AQ n,k is still strongly Menger edge connected even whenthere are 4 n − n ≥ k ≥
3. Moreover, under the restricted condition that eachvertex has at least two fault-free edges, we show that AQ n,k is still strongly Menger edge connectedeven when there are 8 n −
10 faulty edges for n ≥ k ≥
3. These results are all optimal in thesense of the maximum number of tolerated vertex (resp. edge) faults.
Keyword:
Strong Menger (edge) connectivity; Maximal local-connectivity; Augmented k -ary n -cubes; Fault-tolerance. With continuous advances in technology, a multiprocessor system may contains hundreds or even thou-sands of processors that communicate by exchanging messages through an interconnection network. Thetopology of a network can be represented as a graph. Among all fundamental properties for intercon-nection networks, the connectivity and edge connectivity are the major parameters widely discussed forthe connection status of networks.For a connected graph G , the connectivity κ ( G ) is the minimum number of vertices removed toget the graph disconnected or trivial; while the edge connectivity λ ( G ) is the minimum number ofedges removed to get the graph disconnected. Connectivity and edge connectivity are two deterministicmeasurements for determining the reliability and fault tolerance of a multiprocessor system. In contrastwith this concept, Menger [11] provided a local point of view, and defined the connectivity (resp. edgeconnectivity) of any two vertices as the minimum number of internally vertex-disjoint (resp. edge-disjoint)paths between them.A connected graph G is called strongly Menger (edge) connected if for any two distinct vertices x, y of G , there are min { deg G ( x ) , deg G ( y ) } (edge-)disjoint paths between x and y . Parallel routing(i.e., construction of disjoint paths or edge-disjoint paths) has been an important issue in the study of ∗ This work was partially supported by China Postdoctoral Science Foundation (2018M631322). † This work was supported by the grant MOST-107-2221-E-141-001-MY3 from the Ministry of Science and Technology,Taiwan. ‡ This work was partially supported by the National Natural Science Foundation of China (No. 11971054, 11731002)and the 111 Project of China (B16002).
E-mail address : [email protected] (M.-M. Gu), [email protected] (J.-M. Chang), [email protected] (R.-X. Hao) a r X i v : . [ m a t h . C O ] O c t omputer networks. With the continuous increasing in network size, routing in networks with faultshas become unavoidable. Two fault models have been studied for many well-known networks: one isthe random fault model, and the other is the conditional fault model (which assumes that the faultdistribution is limited). The strong Menger (edge) connectivity of a graph with random faults is definedas follows. Definition 1.
A graph G is called m -strongly Menger connected (resp. edge connected ) if G − F remainsstrongly Menger connected (resp. edge connected) for an arbitrary vertex set F ⊆ V ( G ) (resp. edge set F ⊆ E ( G )) with | F | ≤ m .Note that the term m -strong Menger connectivity is also referred to as m -fault-tolerant maximallocal-connectivity in [1,2,15]. Let δ ( G ) denote the minimum degree of a graph G . The conditional strongMenger (edge) connectivity of a graph is defined as follows. Definition 2.
A graph G is called m -conditional strongly Menger connected (resp. edge connected ) if G − F remains strongly Menger connected (resp. edge connected) for an arbitrary vertex set F ⊆ V ( G )(resp. edge set F ⊆ E ( G )) with | F | ≤ m and δ ( G − F ) ≥ k -ary n -cube is proposed for parallel computing by Xiang andStewart [19]. An augmented k -ary n -cube AQ n,k is extended from a k -ary n -cube Q kn in a manneranalogous to the extension of an n -dimensional hypercube Q n to an n -dimensional augmented AQ n [4]and has many triangles. Some results about topological properties and routing problems on augmented k -ary n -cube can be found in [5, 10, 19, 20] etc. In this paper, by exploring and utilizing the structuralproperties of AQ n,k , we show that AQ n, (resp. AQ n,k , k ≥
4) is (4 n − n − n ≥ n ≥ AQ n,k is (4 n − n ≥ k ≥
3. Moreover, under the restricted condition that each vertex has at least two fault-freeedges, we show that AQ n,k is (8 n − n ≥ k ≥ Let G = ( V ( G ) , E ( G )) represent an interconnection network, where a vertex u ∈ V ( G ) represents aprocessor and an edge ( u, v ) ∈ E ( G ) represents a link between vertices u and v .Let | V ( G ) | be the size of vertex set and | E ( G ) | be the size of edge set. Two vertices u and v are adjacent if ( u, v ) ∈ E ( G ), the vertex u is called a neighbor of v , and vice versa. For a vertex u ∈ V ( G ),let N G ( u ) denote a set of vertices in G adjacent to u , and let N G [ u ] = N G ( u ) ∪ { u } . The degree of u ,denoted by deg G ( u ) (or d G ( u )), is the cardinality of N G ( u ). For a vertex set U ⊆ V ( G ), the neighborhood U in G is defined as N G ( U ) = (cid:83) v ∈ U N G ( v ) − U . When no ambiguity arises, we omit the subscript G in the above notations. For any two vertices u, v ∈ V ( G ), we use cn ( G : u, v ) to denote the number ofcommon neighbors of u and v in G .A graph H is a subgraph of a graph G if V ( H ) ⊆ V ( G ) and E ( H ) ⊆ E ( G ). The connected components (simply, component ) of a graph are its maximal connected subgraphs. For two disjoint vertex sets orsubgraphs H and H , we use E ( H , H ) to denote the set of edges with one endpoint in H and theother in H . For a subset S ⊆ V ( G ) (resp. S ⊆ E ( G )), we denote G − S the graph obtained from G byremoving the vertices (edges) of S . In particular, S is called a vertex cut (resp. edge cut ) of G if G − S isdisconnected. In this case, the biggest component of G − S is called a large component ; the componentof G − S which is not the biggest one is called a smaller component .Given x, y ∈ V ( G ), an ( x, y ) -path of length k is a finite sequence of distinct vertices v , v , . . . , v k suchthat x = v , y = v k , and ( v i , v i +1 ) ∈ E ( G ) for 0 ≤ i ≤ k −
1. A set F ⊂ V ( G ) \ { x, y } is an ( x, y ) -cut if G − F has no ( x, y )-path. Similarly, a set F ⊆ E ( G ) is an ( x, y ) -edge cut if G − F has no ( x, y )-path. Proposition 1. ( [11])
Let x, y be two distinct vertices of a graph G . (1) For ( x, y ) / ∈ E ( G ) , the minimum size of an ( x, y ) -cut equals the maximum number of disjoint ( x, y ) -paths. (2) The minimum size of an ( x, y ) -edge cut equals the maximum number of edge-disjoint x, y -paths. k -ary n -cubes Let [ n ] = { , , . . . , n } and [ n ] = { , , . . . , n − } . Assume that all arithmetics on tuple elements aremodulo k . Xiang and Stewart [19] gave two equivalent definitions of augmented k -ary n -cube as follows. Definition 3. ( [19]) Let n ≥ k ≥ augmented k -ary n -cube AQ n,k has k n vertices, each vertex is labelled by an n -bit string ( a n , a n − , . . . , a , a ) (or a n a n − · · · a a ) with a i ∈ [ k ] for i ∈ [ n ]. There is an edge joining vertex u = ( u n , u n − , . . . , u , u ) to v = ( v n , v n − , . . . , v , v ) if andonly if one of the following conditions holds.(1) v i = u i − v i = u i + 1) for some i ∈ [ n ] and u j = v j for all j ∈ [ n ] \ { i } ; and the edge ( u, v )is called an ( i, − -edge (resp. ( i, +1) -edge ).(2) for some 2 ≤ i ≤ n , v i = u i − v i − = u i − − v = u − v i = u i +1, v i − = u i − +1,. . . , v = u + 1), v j = u j for all j > i ; and the edge ( u, v ) is called an ( ≤ i, − -edge (resp. ( ≤ i, +1) -edge ).In the above definition, edges fulfilled the condition (1) and condition (2) are called traditional edges and augmented edges , respectively. Obviously, AQ ,k is a k -cycle (i.e., a cycle of length k ). Fig. 1 shows AQ , , AQ , and AQ , , where bold lines are traditional edges and dashed lines are augmented edges.In fact, the augmented k -ary n -cube AQ n,k can also be recursively defined as follows. Definition 4. ( [19]) Fix k ≥
3, augmented k -ary 1-cube AQ ,k has vertex set { , , . . . , k } , and there isan edge joining vertex u to v if and only if v = u − v = u + 1. Fix n ≥
2. Take k copies of augmented k -ary ( n − AQ n − ,k and for i th copy add an extra number i as the n th bit of each vertex (i.e., allvertices have the same n th bit if they are in the same copy of augmented k -ary ( n − n, − n, +1)-edge, ( ≤ n, − ≤ n, +1)-edge(as defined in Definition 3). Lemma 1. ( [19])
Let AQ n,k be the augmented k -ary n -cube, where n ≥ and k ≥ are integers. (1) For i ∈ [ k ] , the subgraph of AQ n,k induced by vertices with the n th bit being i , denoted by AQ in,k ,is a copy of AQ n − ,k .
Let u and v be two distinct vertices in AQ in,k such that u and v have common neighborsin V ( AQ i +1 n,k ) ∪ V ( AQ i − n,k ) . Then v = u ( ≤ n − , − or v = u ( ≤ n − , +1) . Furthermore, if v = u ( ≤ n − , − , thenthe common neighbors of u and v in V ( AQ i +1 n,k ) ∪ V ( AQ i − n,k ) are u ( ≤ n, − and u ( n, +1) ; if v = u ( ≤ n − , +1) ,then the common neighbors of u and v in V ( AQ i +1 n,k ) ∪ V ( AQ i − n,k ) are u ( ≤ n, +1) and u ( n, − . For instance, we consider u = 120 in AQ , (see Fig. 1(c)). Then, we can check that u and u ( ≤ , − =412 have common neighbors u ( ≤ , − = 012 ∈ V ( AQ , ) and u (3 , +1) = 220 ∈ V ( AQ , ). Also, u and u ( ≤ , +1) = 101 have common neighbors u ( ≤ , +1) = 201 ∈ V ( AQ , ) and u (3 , − = 020 ∈ V ( AQ , ).The following lemma shows the exact number of common neighbors of any two adjacent vertices in AQ n,k . Lemma 2. ( [10])
Let ( u, v ) be an edge of AQ n,k . Then the following assertions hold: (1) cn ( AQ , : u, v ) = 3 ; (2) For k ≥ , cn ( AQ ,k : u, v ) = 2 ; (3) For n ≥ , cn ( AQ n, : u, v ) = (cid:40) if v = u ( ≤ n, − or v = u ( i, − for ≤ i ≤ n ;5 if v = u ( ≤ i, − for ≤ i ≤ n − For n ≥ and k ≥ , cn ( AQ n,k : u, v ) = (cid:40) if v = u ( ≤ n, − or v = u ( i, − for ≤ i ≤ n ;4 if v = u ( ≤ i, − for ≤ i ≤ n − . The results are similar with assertions (3) and (4) for v = u ( i, +1) and v = u ( ≤ i, +1) . The following lemma shows the upper bound of the number of common neighbors for any two distinctvertices in AQ n,k . Lemma 3. ( [10])
Let u and v be two distinct vertices of AQ n,k . Then the following assertions hold: (1) For n ≥ , cn ( AQ n, : u, v ) ≤ ; (2) For k ≥ , cn ( AQ ,k : u, v ) ≤ ; (3) For n ≥ and k ≥ , cn ( AQ n,k : u, v ) ≤ . Lemma 4. ( [5])
Let AQ n,k be the augmented k -ary n -cubes, where n ≥ and k ≥ . If U is a subsetof V ( AQ n,k ) with ≤ | U | ≤ n − , then | N AQ n,k ( U ) | ≥ n − . Wang and Zhao [18] derived the following result which is useful for our proof.
Lemma 5. ( [18])
Let AQ n, be the augmented -ary n -cubes, and F ⊆ V ( AQ n, ) with | F | ≤ n − .Assume AQ n, − F is disconnected. Then (1) AQ , − F has exactly two components, one of which is a singleton or a -cycle, and the vertex setof the -cycle is { u, u ( ≤ , − , u ( ≤ , +1) } ; (2) for n ≥ , AQ n, − F has exactly two components, one of which is a singleton. In this section, we will consider the (conditional) fault-tolerant strong Menger (edge) connectivity ofaugmented k -ary n -cubes. The following result is useful. Lemma 6. ( [15])
Let G be an r -regular, r -connected graph with | V ( G ) | ≥ r + 1 and r ≥ . Then G is f -strongly Menger connected if, for any V f ⊆ V ( G ) with | V f | ≤ f + r − , G − V f has a component C such that | V ( C ) | ≥ | V ( G ) | − | V f | − . k -ary n -cubes In this section, we consider the strong Menger connectivity of augmented k -ary n -cubes AQ n,k . We willprove that AQ n, is (4 n − n − n ≥
3, and AQ n,k is (4 n − n − n ≥ k ≥
4. 5 emma 7.
Let F be an arbitrary set of vertices in AQ n,k . (1) If | F | ≤ n − for n ≥ , then AQ n, − F has a component C such that | V ( C ) | ≥ | V ( AQ n, ) | −| F | − . (2) If | F | ≤ n − for n ≥ and k ≥ , then AQ n,k − F has a component C such that | V ( C ) | ≥| V ( AQ n,k ) | − | F | − . Proof.
Let C be the large component of AQ n,k − F . First we consider k = 3, n ≥ | F | ≤ n − AQ n, − F is connected or has exactly two components, one of which is a singleton.Clearly, if AQ n,k − F is connected, then V ( C ) = V ( AQ n,k − F ) and | V ( C ) | = | V ( AQ n, ) | − | F | . If AQ n, − F is disconnected, then | V ( C ) | = | V ( AQ n, ) | − | F | − n ≥ k ≥ | F | ≤ n −
11. The proof is by induction on n . If n = 2,then | F | ≤ < κ ( AQ ,k ) = 6. Thus, AQ ,k − F is connected. It leads to V ( C ) = V ( AQ ,k − F ) and | V ( C ) | = | V ( AQ ,k ) | − | F | . In what follows, we assume that n ≥ AQ n − ,k .Recall that AQ n,k contains k disjoint copies of AQ n − ,k , say AQ in,k , i ∈ [ k ] . Let F i = F ∩ V ( AQ in,k )and f i = | F i | for i ∈ [ k ] . Let I = { i ∈ [ k ] : AQ in,k − F i is disconnected } and J = [ k ] \ I . In addition,we adopt the following notations: F I = (cid:91) i ∈ I F i , F J = (cid:91) j ∈ J F j , and AQ Jn,k = (cid:91) j ∈ J AQ jn,k . By Lemma 1(2), f i ≥ n − − n − i ∈ I . Since | F | ≤ n −
11, we have | I | ≤
2. We considerthe following cases.Case 1: | I | = 0.For all j ∈ [ k ] , AQ jn,k − F j is connected. By Lemma 1(3), there are 2 k n − edges between subgraphs AQ jn,k and AQ j +1 n,k . Since | F | ≤ n − < k n − for n ≥ k ≥
4, there is a fault-free edgebetween AQ jn,k − F j and AQ j +1 n,k − F j +1 for each j ∈ [ k ] , it implies that AQ n,k − F is connected. Let C = AQ n,k − F . Clearly, | V ( C ) | = | V ( AQ n,k ) | − | F | .Case 2: | I | = 1.Without loss of generality, assume that I = { } . By Lemma 1(2), f ≥ n −
6. For j ∈ [ k ] \ { } , AQ jn,k − S j is connected. By Lemma 1(3), there are 2 k n − edges between subgraphs AQ jn,k and AQ j +1 n,k .For each j, j + 1 ∈ [ k ] \ I , since 2( f j + f j +1 ) ≤ | F | ≤ n − < k n − for n ≥ k ≥
4, there isa fault-free edge between AQ jn,k − F j and AQ j +1 n,k − F j +1 . It leads to AQ Jn,k − F J is connected.Case 2.1: 4 n − ≤ f ≤ n − f ≤ n −
19 = 8( n − −
11, by induction hypothesis on AQ n,k , there exists a component, say H in AQ n,k , such that | V ( H ) | ≥ | V ( AQ n,k ) |− f − k n − − f −
1. Since there are 2 × k n − edges betweensubgraphs AQ n,k and AQ Jn,k and 2( f + f + 1) + 2( f + f k − + 1) ≤ | F | + 4 ≤ n −
11) + 4 < × k n − for n ≥ k ≥ H is connected to AQ Jn,k − F J . Let C be the component induced by the vertex set V ( H ) ∪ V ( AQ Jn,k − F J ). Then | V ( C ) | ≥ | V ( AQ n,k ) | − | F | − n − ≤ f ≤ n − | F J | = | F | − f ≤
7. Let H be the large component of AQ n,k − F and M = AQ n,k − F − V ( H ). Obviously, N AQ n,k − AQ n,k ( V ( M )) ⊆ F J and N AQ n,k V ( M ) ⊆ F . By Lemma 1(4), | N AQ n,k − AQ n,k V ( M ) | ≥ | V ( M ) | . It leads to 2 | V ( M ) | ≤ | F J | ≤
7, so | V ( M ) | ≤
3. If 2 ≤ | V ( M ) | ≤ | N AQ n,k ( V ( M )) | ≥ n −
10. It leads to8 n − ≥ | F | = f + | F J | ≥ | N AQ n,k V ( M ) | + | N AQ n,k − AQ n,k ( V ( M )) | = | N AQ n,k ( V ( M )) | ≥ n − , a contradiction. Thus, | V ( M ) | = 1 and | V ( H ) | = | V ( AQ n,k ) | − f −
1. One can see that H is connectedto AQ Jn,k − F J by the similar argument as Case 2.1. Let C be the component induced by the vertex set V ( H ) ∪ V ( AQ Jn,k − F J ). Then | V ( C ) | ≥ | V ( AQ n,k ) | − | F | − | I | = 2.Without loss of generality, assume that I = { , t } , where 1 ≤ t ≤ k −
1. By Lemma 1(2), f , f t ≥ n − | F | ≤ n −
11, we have | F J | ≤
1. For j ∈ [ k ] \ { , t } , AQ jn,k − F j is connected. We consider thefollowing cases.Case 3.1: t = 1 or k − t = 1. Note that | F J | ≤ k n − > | F J | for n ≥ j, j + 1 ∈ [ k ] \ { , } , there is a fault-free edge between AQ jn,k − F j and AQ j +1 n,k − F j +1 . It leadsto AQ Jn,k − F J is connected. Since every vertex of AQ n,k (resp. AQ n,k ) has two extra neighbors in AQ Jn,k and | F J | ≤
1, any component of AQ n,k − F (resp. AQ n,k − F ) is connected to AQ Jn,k − F J . Let C = AQ n,k − F . Clearly, | V ( C ) | = | V ( AQ n,k ) | − | F | .Case 3.2: 2 ≤ t ≤ k − ≤ m ≤ t − t + 1 ≤ m ≤ k − AQ mn,k − F m is connected. By the similar argument asCase 3.1, those ( AQ mn,k − F m )’s for 1 ≤ m ≤ t − t +1 ≤ m ≤ k − C and C , respectively, of AQ n,k − F . Since every vertex of AQ n,k (resp. AQ tn,k ) has four extra neighborsin AQ n,k − AQ n,k − AQ tn,k and | F J | ≤
1, any component of AQ n,k − F is connected to both AQ n,k − F (which is part of C ) and AQ k − n,k − F k − (which is part of C ). This implies that C = C (i.e., C and C are the same component). By a similar discussion, any component of AQ tn,k − F t is contained in both C and C . This implies that AQ n,k − F has a large component, say C , and C = C = C . It leads to | V ( C ) | = | V ( AQ n,k ) | − | F | . (cid:3) Since AQ n,k is (4 n − n − Theorem 1.
Let AQ n,k be the augmented k -ary n -cube. Then (1) AQ n, is (4 n − -strongly Menger edge connected for n ≥ . (2) AQ n,k is (4 n − -strongly Menger edge connected for n ≥ and k ≥ . Remark 2. AQ n, is not (4 n − n ≥ AQ n,k is not (4 n − n ≥ k ≥
4. See Fig. 2 for an illustration. Let ( u, w ) ∈ E ( AQ n,k ) such that u = w ( ≤ i, − for some 1 ≤ i ≤ n − F = N ( w ) \ N [ u ] be a faulty subset of vertices in AQ n,k (i.e., vertices in the darkest area of Fig. 2). By Lemma 2(3), | F | = (4 n − − − n − n ≥ k = 3, and by Lemma 2(4), | F | = (4 n − − − n − n ≥ k ≥
4. We now considera vertex v ∈ V ( AQ n,k ) \ ( N [ N ( w ) ∪ N ( u )]). Obviously, deg AQ n,k − F ( u ) = deg AQ n,k − F ( v ) = 4 n − w ∈ N ( u ) and some neighbors of w are in F , the vertices u and v are not connected with 4 n − AQ n,k − F . Thus, the results of Theorem 1 are optimal in the sense that thenumber of faulty vertices cannot be increased. k -ary n -cubes In this section, we consider the (conditional) strongly Menger edge connectivity of augmented k -ary n -cubes.In the following, let S be an arbitrary set of edges in AQ n,k . Note that AQ n,k contains k disjointcopies of AQ n − ,k , say AQ in,k , i ∈ [ k ] . Let S i = S ∩ E ( AQ in,k ) and s i = | S i | for i ∈ [ k ] . Let I = { i ∈ [ k ] : AQ in,k − S i is disconnected } and J = [ k ] \ I . In addition, we adopt the followingnotations: S I = (cid:91) i ∈ I S i , S J = (cid:91) j ∈ J S j , AQ Jn,k = (cid:91) j ∈ J AQ jn,k , and s c = | S | − (cid:88) i ∈ [ k ] s i . ( w )
Lemma 8.
Let S be an arbitrary set of edges in AQ n,k for n ≥ and k ≥ . If | S | ≤ n − , then thereexists a component H in AQ n,k − S such that | V ( H ) | ≥ | V ( AQ n,k ) | − . Proof.
Let H be the large component of AQ n,k − S . The proof is by induction on n . For n = 2, theproof is shown in the Appendix A. In what follows, we assume that n ≥ k ≥ AQ n − ,k . Recall that I = { i ∈ [ k ] : AQ in,k − S i is disconnected } and J = [ k ] \ I . By Lemma 1(2), s i ≥ n − − n − i ∈ I . Since | S | ≤ n −
7, we have | I | ≤ n ≥
3. The followingcases should be considered.Case 1: | I | = 0.For all j ∈ [ k ] , AQ jn,k − S j is connected. By Lemma 1(3), there are 2 k n − edges between subgraphs AQ jn,k and AQ j +1 n,k . Since | S | ≤ n − < k n − for n ≥ k ≥
3, there is a fault-free edgebetween AQ jn,k − S j and AQ j +1 n,k − S j +1 for each j ∈ [ k ] , it implies that AQ n,k − S is connected. Thus, | V ( H ) | = | V ( AQ n,k ) | .Case 2: | I | = 1.Without loss of generality, assume that I = { } . By Lemma 1(2), s ≥ n − − n −
6. For j ∈ [ k ] \ { } , AQ jn,k − S j is connected. By Lemma 1(3), there are 2 k n − edges between subgraphs AQ jn,k and AQ j +1 n,k . Since s c ≤ | S | − s ≤ (8 n − − (4 n −
6) = 4 n − < k n − for n ≥ k ≥
3, there is afault-free edge between AQ jn,k − S j and AQ j +1 n,k − S j +1 for each j, j + 1 ∈ J . It leads to AQ Jn,k − S J isconnected.Case 2.1: 4 n − ≤ s ≤ n − s ≤ n −
15 = 8( n − −
7, by induction hypothesis on AQ n,k , there exists a component, say H in AQ n,k , such that | V ( H ) | ≥ | V ( AQ n,k ) | − k n − −
1. Since every vertex of H has exactly fourdistinct extra neighbors, we have s c ≤ | S |− s ≤ n − < × ( k n − −
1) for n ≥ k ≥
3, and thus H is connected to AQ Jn,k − S J . Let H be the component induced by the vertex set V ( H ) ∪ V ( AQ Jn,k − S J ).Then | V ( H ) | ≥ | V ( AQ n,k ) | −
1, as desired.Case 2.2: 8 n − ≤ s ≤ n − s c ≤ | S | − s ≤ (8 n − − (8 n −
14) = 7. Since every vertex in AQ n,k hasexactly four distinct extra neighbors, at most one vertex in AQ n,k are not connected with AQ Jn,k − S J .This shows that | V ( H ) | ≥ | V ( AQ n,k ) | −
1, as desired.Case 3: | I | = 2. 8e consider the following two cases according to k = 3 or not.Case 3.1: k = 3.Without loss of generality, assume that I = { , } and s ≤ s . Then, AQ Jn, − S J = AQ n, − S isconnected. Since | S | ≤ n −
7, if s ≥ n −
14, then s c ≤ | S | − s − s ≤ (8 n − − n −
14) = 21 − n ,which contradicts that n ≥
3. Indeed, by Lemma 1(2), we have 4 n − ≤ s ≤ s ≤ | S |− (4 n − ≤ n − s c ≤ | S | − s − s ≤ (8 n − − n −
6) = 5. Thus, we consider the following two situations.Case 3.1.1: 4 n − ≤ s ≤ s ≤ n −
15 = 8( n − − AQ n, − S (resp. AQ n, − S ) is disconnected. By induction hypothesis on AQ n, (resp. AQ n, ), AQ n, − S (resp. AQ n, − S ) has a large component and a singleton, say x (resp. x ). Notethat each singleton has exactly two distinct extra neighbors in AQ n, . Since s c ≤ < k n − − n ≥ k = 3, AQ n, − S − { x } (resp. AQ n, − S − { x } ) is connected to AQ n, − S . Let M be the union of smaller components of AQ n, − S . Clearly, V ( M ) ⊆ { x , x } . If | V ( M ) | = 2, then s c ≥ | N AQ n, ∪ AQ n, ( x ) | + | N AQ n, ( x ) | ≥
6, a contradiction occurs. Thus, | V ( M ) | ≤
1. This impliesthat AQ n, − S has a large component H and smaller components which contain at most one vertices intotal. It leads to | V ( H ) | ≥ | V ( AQ n, ) | − n − ≤ s ≤ n −
15 and 8 n − ≤ s ≤ n − n −
6) + (8 n − ≤ s + s ≤ | S | ≤ n −
7. It implies that n = 3. Thus, wehave | S | ≤
17, 6 ≤ s ≤
9, 10 ≤ s ≤
11 and s c ≤ | S | − s − s ≤
1. Since every vertex in AQ n, (resp. AQ n, ) has two distinct extra neighbors in AQ n, , any component in AQ n, − S (resp. AQ n, − S ) isconnected to AQ n, − S . This implies AQ n, − S is connected and | V ( H ) | = | V ( AQ n, ) | .Case 3.2: k ≥ ∈ I . We consider the following two cases according to thevalue of I \ { } .Case 3.2.1: I = { , } or I = { , k − } .Without loss of generality, assume that I = { , } and s ≤ s . For j ∈ [ k ] \ { , } , AQ jn,k − S j isconnected. By Lemma 1(3), since s c ≤ | S | − s − s ≤ (8 n − − n −
6) = 5 < k n − for n ≥ k ≥
4, there exists a fault-free edge joining AQ jn,k − S j and AQ j +1 n,k − S j +1 for j, j + 1 ∈ [ k ] \ { , , k − } .It leads to AQ Jn,k − S J is connected. By Lemma 1(2), since | S | ≤ n −
7, we have 4 n − ≤ s ≤ s ≤| S | − (4 n − ≤ n −
1. The following two cases should be considered.Case 3.2.1a: 4 n − ≤ s ≤ s ≤ n − n − ≤ s ≤ n −
15 and 8 n − ≤ s ≤ n − I = { , t } , where 2 ≤ t ≤ k − ≤ m ≤ t − t + 1 ≤ m ≤ k − AQ mn,k − S m is connected. By the similar argument asCase 3.2.1, those ( AQ mn,k − S m )’s for 1 ≤ m ≤ t − t + 1 ≤ m ≤ k − C and C , respectively, of AQ n,k − S . Without loss of generality, assume that s ≤ s t . We considerthe following cases.Case 3.2.2a: 4 n − ≤ s ≤ s t ≤ n −
15 = 8( n − − AQ n,k − S (resp. AQ tn,k − S t ) is disconnected. By induction hypothesis on AQ n,k (resp. AQ tn,k ), AQ n,k − S (resp. AQ tn,k − S t ) has exactly two components: a large component, say H (resp. H t ), and a singleton. Note that every vertex in AQ n,k has exactly two distinct extra neighbors in AQ n,k (resp. AQ k − n,k ). Since s c ≤ | S | − s − s t ≤ < k n − − n ≥ k ≥
4, there is at least a fault-free9dge between H and AQ n,k − S (resp. AQ k − n,k − S k − ). This implies that H is connected to AQ n,k − S (which is part of C ) and is connected to AQ k − n,k − S k − (which is part of C ). It follows that H iscontained in both C and C . By a similar discussion, H t is contained in both C and C .Let M be the union of smaller components of AQ n,k − S . Clearly, V ( M ) ⊆ { x , x t } . If | V ( M ) | = 2,then s c ≥ | N AQ n,k − AQ n,k ( x ) | + | N AQ n,k − AQ tn,k ( x t ) | ≥
8, a contradiction occurs. Thus, | V ( M ) | ≤ AQ n,k − S has a large component H and smaller components which contain at mostone vertices in total. So | V ( H ) | ≥ | V ( AQ n,k ) | − n − ≤ s ≤ n −
15 and 8 n − ≤ s t ≤ n − n −
6) + (8 n − ≤ s + s t ≤ | S | ≤ n −
7. It implies that n = 3. Thus, | S | ≤
17, 6 ≤ s ≤
9, 10 ≤ s t ≤
11 and s c ≤ | S | − s − s t ≤
1. Since every vertex in AQ n,k (resp. AQ tn,k ) has four distinct extra neighbors in AQ n,k − AQ n,k − AQ tn,k , any component in AQ n,k − S (resp. AQ tn,k − S t ) is connected to AQ n,k − S (resp. AQ t − n,k − S t − ) (which is part of C ) and is connectedto AQ k − n,k − S k − (resp. AQ t +1 n,k − S t +1 ) (which is part of C ). This implies AQ n,k − S is connected andthus | V ( H ) | = | V ( AQ n,k ) | . (cid:3) Lemma 9.
Let S be an arbitrary set of edges in AQ n,k for n ≥ and k ≥ . If | S | ≤ n − , thenthere exists a component H in AQ n,k − S such that | V ( H ) | ≥ | V ( AQ n,k ) | − . Proof.
Let H be the large component of AQ n,k − S . The proof is by induction on n . For n = 2, theproof is shown in the Appendix B. In what follows, we assume that n ≥ k ≥ AQ n − ,k . Recall that I = { i ∈ [ k ] : AQ in,k − S i is disconnected } and J = [ k ] \ I . By Lemma 1(2), s i ≥ n − − n − i ∈ I . Since | S | ≤ n −
13, we have | I | ≤ n ≥
3. We consider thefollowing cases.Case 1: | I | = 0.For all j ∈ [ k ] , AQ jn,k − S j is connected. By Lemma 1(3), there are 2 k n − edges between subgraphs AQ jn,k and AQ j +1 n,k . Since | S | ≤ n − < × (2 k n − ) for n ≥ k ≥
3, there exists at mostone integer, say i ∈ [ k ] , such that all the edges between AQ in,k and AQ i +1 n,k are faulty. Since there is afault-free edge between AQ jn,k − S j and AQ j +1 n,k − S j +1 for each j ∈ [ k ] \ { i } , it implies that AQ n,k − S is connected. Thus, | V ( H ) | = | V ( AQ n,k ) | .Case 2: | I | = 1.Without loss of generality, assume that I = { } . By Lemma 1(2), s ≥ n − − n −
6. For j ∈ [ k ] \ { } , AQ jn,k − S j is connected. By Lemma 1(3), there are 2 k n − edges between subgraphs AQ jn,k and AQ j +1 n,k . Since s c ≤ | S | − s ≤ (12 n − − (4 n −
6) = 8 n − < k n − for n ≥ k ≥
3, there isa fault-free edge between AQ jn,k − S j and AQ j +1 n,k − S j +1 for each j, j + 1 ∈ J . It leads to AQ Jn,k − S J isconnected.Case 2.1: 4 n − ≤ s ≤ n − s ≤ n −
25 = 12( n − −
13, by induction hypothesis on AQ n,k , there exists a component, say H in AQ n,k , such that | V ( H ) | ≥ | V ( AQ n,k ) | − k n − −
2. Since every vertex of H has exactly fourdistinct extra neighbors, we have s c ≤ | S |− s ≤ n − < × ( k n − −
2) for n ≥ k ≥
3, and thus H is connected to AQ Jn,k − S J . Let H be the component induced by the vertex set V ( H ) ∪ V ( AQ Jn,k − S J ).Then | V ( H ) | ≥ | V ( AQ n,k ) | − n − ≤ s ≤ n − s c ≤ | S | − s ≤ (12 n − − (12 n −
24) = 9. Since every vertex in AQ n,k hasexactly four distinct extra neighbors, at most two vertices in AQ n,k are not connected with AQ Jn,k − S J .This shows that | V ( H ) | ≥ | V ( AQ n,k ) | −
2, as desired.10ase 3: | I | = 2.We consider the following two cases according to k = 3 or not.Case 3.1: k = 3.Without loss of generality, assume that I = { , } and s ≤ s . Then, AQ Jn, − S J = AQ n, − S isconnected. By Lemma 1(2), since | S | ≤ n −
13, we have 4 n − ≤ s ≤ s ≤ | S | − (4 n − ≤ n − n − ≤ s ≤ s ≤ n −
15 = 8( n − − AQ n, − S (resp. AQ n, − S ) is disconnected. By Lemma 8, AQ n, − S (resp. AQ n, − S )has a large component and a singleton, say x (resp. x ). Note that the singleton has exactly two distinctextra neighbors in AQ n, . Since s c ≤ | S | − s − s ≤ (12 n − − n −
6) = 4 n − < k n − − n ≥ k = 3, AQ n, − S − { x } (resp. AQ n, − S − { x } ) is connected to AQ n, − S . This impliesthat AQ n, − S has a large component H and smaller components which contain at most two vertices intotal. It leads to | V ( H ) | ≥ | V ( AQ n,k ) | − n − ≤ s ≤ n −
15 and 8 n − ≤ s ≤ n − s c ≤ | S | − s − s ≤ (12 n − − (4 n − − (8 n −
14) = 7. Since every vertexin AQ n, (resp. AQ n, ) has four distinct extra neighbors, any component with more than two vertices in AQ n, − S (resp. AQ n, − S ) is connected to AQ n, − S . This implies that only a component with asingleton in AQ n, − S (resp. AQ n, − S ) can be disconnected with AQ n, − S . Thus, AQ n, − S hasa large component H and smaller components which contain at most two vertices in total. It leads to | V ( H ) | ≥ | V ( AQ n, ) | − n − ≤ s ≤ s ≤ n − n − ≤ s + s ≤ | S | ≤ n −
13. It implies that n = 3. Thus, | S | ≤ n −
13 = 23 and 10 ≤ s ≤ s ≤
17. Also, we have s c ≤ | S | − s − s ≤ − ×
10 = 3. Notethat every vertex in AQ n, (resp. AQ n, ) has two distinct extra neighbors in AQ n, , at most one vertexin ( AQ n, − S ) ∪ ( AQ n, − S ) can be disconnected with AQ n, − S . If a vertex v in AQ n, (resp. AQ n, ) remains a singleton in AQ n, − F , then all the extra edges incident with v are in S c . It impliesthat s c ≥
4, a contradiction. Thus, any component of AQ n, − S (resp. AQ n, − S ) is connected to AQ n, − S . It leads to that AQ n, − S is connected, and so | V ( H ) | = | V ( AQ n, ) | .Case 3.2: k ≥ ∈ I . We consider the following two cases according to thevalue of I \ { } .Case 3.2.1: I = { , } or I = { , k − } .Without loss of generality, assume that I = { , } and s ≤ s . For j ∈ [ k ] \ { , } , AQ jn,k − S j isconnected. By Lemma 1(3), since s c ≤ | S | − s − s ≤ (12 n − − n −
6) = 4 n − < k n − for n ≥ k ≥
4, there exists a fault-free edge joining AQ jn,k − S j and AQ j +1 n,k − S j +1 for j ∈ [ k ] \ { , , k − } .It leads to AQ Jn,k − S J is connected. By Lemma 1(2), since | S | ≤ n −
13, we have 4 n − ≤ s ≤ s ≤| S | − (4 n − ≤ n −
7. The following three cases should be considered.Case 3.2.1a: 4 n − ≤ s ≤ s ≤ n − n − ≤ s ≤ n −
15 and 8 n − ≤ s ≤ n − n − ≤ s ≤ s ≤ n − I = { , t } , where 2 ≤ t ≤ k −
2. 11or 1 ≤ m ≤ t − t + 1 ≤ m ≤ k − AQ mn,k − S m is connected. By the similar argument asCase 3.2.1, those ( AQ mn,k − S m )’s for 1 ≤ m ≤ t − t + 1 ≤ m ≤ k − C and C , respectively, of AQ n,k − S . Without loss of generality, assume that s ≤ s t . We considerthe following cases.Case 3.2.2a: 4 n − ≤ s ≤ s t ≤ n −
15 = 8( n − − AQ n,k − S (resp. AQ tn,k − S t ) is disconnected. By Lemma 8, AQ n,k − S (resp. AQ tn,k − S t ) has exactly two components: a large component, say H (resp. H t ), and a singleton. Notethat every vertex in AQ n,k has exactly two distinct extra neighbors in AQ n,k (resp. AQ k − n,k ). Since s c ≤ | S | − s − s t ≤ (12 n − − n −
6) = 4 n − < k n − − n ≥ k ≥
4, there is at least afault-free edge between H and AQ n,k − S (resp. AQ k − n,k − S k − ). This implies that H is connected to AQ n,k − S (which is part of C ) and is connected to AQ k − n,k − S k − (which is part of C ). It follows that H is contained in both C and C . By a similar discussion, H t is contained in both C and C . Thisimplies that AQ n,k − S has a large component H = C = C and smaller components which contain atmost two vertices in total. It leads to | V ( H ) | ≥ | V ( AQ n,k ) | − n − ≤ s ≤ n −
15 and 8 n − ≤ s t ≤ n − s c ≤ | S | − s − s t ≤ (12 n − − (4 n − − (8 n −
14) = 7. Since every vertexin AQ n,k (resp. AQ tn,k ) has four distinct extra neighbors in AQ n,k − AQ n,k − AQ tn,k , any componentwith more than two vertices in AQ n,k − S is connected to AQ n,k − S (which is part of C ) and isconnected to AQ k − n,k − S k − (which is part of C ). This implies that only a component with a singletonin AQ n,k − S can be disconnected with both C and C . By a similar discussion, only a component witha singleton in AQ tn,k − S t can be disconnected with both C and C . This implies that AQ n,k − S has alarge component H = C = C and smaller components which contain at most two vertices in total. Itleads to | V ( H ) | ≥ | V ( AQ n,k ) | − n − ≤ s ≤ s t ≤ n − n − ≤ s + s t ≤ | S | ≤ n −
13. It implies that n = 3. Thus, | S | ≤ n −
13 = 23 and 10 ≤ s ≤ s t ≤
17. Also, we have s c ≤ | S | − s − s t ≤ − ×
10 = 3. Sinceevery vertex in AQ n,k has four distinct extra neighbors in AQ n,k − AQ n,k − AQ tn,k , any component of AQ n,k − S is connected to AQ n,k − S (which is part of C ) and is connected to AQ k − n,k − S k − (which ispart of C ). By a similar discussion, any component of AQ tn,k − S t is connected to AQ t − n,k − S t − (whichis part of C ) and is connected to AQ t +1 n,k − S t +1 (which is part of C ). This implies that AQ n,k − S isconnected, and so | V ( H ) | = | V ( AQ n,k ) | .Case 4: | I | = 3.By Lemma 1(2), for each i ∈ I , s i ≥ n − − n − s i ≤ (12 n − − n −
6) = 4 n − < n −
7. Since | S | ≤ n −
13, we have s c ≤ | S |− n −
6) = 5. For each i ∈ I , AQ in,k − S i is disconnected,and by Lemma 8, AQ in,k − S i has two components, one is the large component, say H i , and the other isa singleton, say v i . Let M be the union of smaller components of AQ n,k − S . We consider the followingtwo cases according to k = 3 or not.Case 4.1: k = 3.In this case, I = { , , } and J = ∅ . Since s c ≤ < k n − − n ≥
3, all H i ’s for i ∈ I belongto the same component (i.e., H ) in AQ n, − S . Clearly, V ( M ) ⊆ { v , v , v } . We claim | V ( M ) | ≤ s c ≥ | N AQ n, ∪ AQ n, ( v ) | + | N AQ n, ( v ) | = 4 + 2 = 6, a contradiction. This implies that AQ n, − S has a large component and smaller components which contain at most two vertices in total.It leads to | V ( H ) | ≥ | V ( AQ n,k ) | − k ≥ I are consecutive.Without loss of generality, assume that I = { , , } . For j ∈ [ k ] \ { , , } , AQ jn,k − S j is connected.By Lemma 1(3), there are 2 k n − edges between subgraphs AQ jn,k and AQ j +1 n,k j ∈ [ k ] . Since s c ≤ < k n − − n ≥ k ≥
4, all H i ’s for i ∈ I and all subgraphs ( AQ jn,k − S j )’s for j ∈ J belong tothe same component (i.e., H ) in AQ n,k − S . Clearly, V ( M ) ⊆ { v , v , v } . By the similar discussion asCase 4.1, we can show that | V ( M ) | ≤ | V ( H ) | ≥ | V ( AQ n,k ) | − I are consecutive.Without loss of generality, assume that I = { , , t } , where t ∈ { , , . . . , k − } . For 2 ≤ m ≤ t − t + 1 ≤ m ≤ k − AQ mn,k − S m is connected. By the similar argument as Case 4.2.1, those( AQ mn,k − S m )’s for 2 ≤ m ≤ t − t + 1 ≤ m ≤ k − C and C ,respectively, of AQ n,k − S . Since s c ≤ < k n − − n ≥ k ≥ H is connected to H and AQ k − n,k − S k − (which is part of C ), H is connected to AQ n,k − S (which is part of C ), H t isconnected to AQ t − n,k − S t − (which is part of C ) and is connected to AQ t +1 n,k − S t +1 (which is part of C ). This implies that AQ n,k − S has a large component H = C = C and V ( M ) ⊆ { v , v , v t } . Weclaim | V ( M ) | ≤
2. Otherwise, s c ≥ | N AQ k − n,k ( v ) | + | N AQ t − n,k ∪ AQ t +1 n,k ( v t ) | = 2 + 4 = 6, a contradiction.This implies that AQ n,k − S has a large component H and smaller components which contain at mosttwo vertices in total. It leads to | V ( H ) | ≥ | V ( AQ n,k ) | − I are consecutive.Without loss of generality, suppose I = { , t, p } , where 2 ≤ t < p ≤ k − p − t ≥
2. For1 ≤ m ≤ t − t + 1 ≤ m ≤ p − p + 1 ≤ m ≤ k − AQ mn,k − S m is connected. By thesimilar argument as Case 4.2.1, those ( AQ mn,k − S m )’s for 1 ≤ m ≤ t − t + 1 ≤ m ≤ p −
1, and p + 1 ≤ m ≤ k − C , C and C , respectively, of AQ n,k − S . Since s c ≤ < k n − − n ≥ k ≥ H is connected to AQ n,k − S (which is part of C ) and isconnected to AQ k − n,k − S k − (which is part of C ), H t is connected to AQ t − n,k − S t − (which is part of C )and is connected to AQ t +1 n,k − S t +1 (which is part of C ), H p is connected to AQ p − n,k − S p − (which is partof C ) and is connected to AQ p +1 n,k − S p +1 (which is part of C ). This implies that AQ n,k − S has a largecomponent H = C = C = C and V ( M ) ⊆ { v , v t , v p } . Since 4 | V ( M ) | ≤ s c ≤
5, we have | V ( M ) | ≤ AQ n,k − S has a large component H and smaller component which contain at mostone vertex. It leads to | V ( H ) | ≥ | V ( AQ n,k ) | − (cid:3) Theorem 2.
Let AQ n,k be the augmented k -ary n -cube, where n ≥ and k ≥ are integers. Then AQ n,k is (4 n − -strongly Menger edge connected. Proof.
Let F be an arbitrary faulty edge set of AQ n,k with | F | ≤ n −
4. Since λ ( AQ n,k ) = 4 n − AQ n,k − F is connected. Let u, v ∈ V ( AQ n,k ) be any two distinct vertices such that deg AQ n,k − F ( u ) ≤ deg AQ n,k − F ( v ), and let d u = deg AQ n,k − F ( u ). From Proposition 1, we need to show that the minimumsize of a ( u, v )-edge cut is equal to d u . That is, we will show that u and v are still connected after theremoval of at most d u − AQ n,k − F .Suppose, on the contrary, that u and v are separated by deleting a set of edges E f with | E f | ≤ d u − AQ n,k − F . That is, u is disconnected with v in AQ n,k − ( F ∪ E f ). Since d u = deg AQ n,k − F ( u ) ≤ deg AQ n,k ( u ) = 4 n −
2, we have | E f | ≤ n −
3. Let S = F ∪ E f . Then | S | ≤ (4 n −
4) + (4 n −
3) =8 n −
7. By Lemma 8, there is a component H in AQ n,k − S such that | V ( H ) | ≥ | V ( AQ n,k ) | − | V ( H ) | (cid:54) = | V ( AQ n,k ) | , for otherwise, both u and v are contained in H . Since u is disconnectedfrom v in AQ n,k − S , without loss of generality, assume that u is a singleton in AQ n,k − S . Clearly, E ( { u } , N AQ n,k − F ( u )) ⊆ E f . Thus, | E f | ≥ | N AQ n,k − F ( u ) | = deg AQ n,k − F ( u ) = d u , which contradict to | E f | ≤ d u −
1. This shows that AQ n,k is (4 n − (cid:3) Remark 3.
To show that AQ n,k is not (4 n − u, w ) ∈ E ( AQ n,k ) and v ∈ V ( AQ n,k ) \ N [ w ]. Let F = E ( { w } , N ( w ) \ { u } ) be a faulty subset of edges in AQ n,k (i.e., edges with cross marks in the darkestarea of Fig. 3). Clearly, | F | = 4 n − n − u and v in AQ n,k − F . Since deg AQ n,k − F ( u ) = deg AQ n,k − F ( v ) = 4 n − AQ n,k is not (4 n − N ( u )
Theorem 3.
Let AQ n,k be the augmented k -ary n -cubes, where n ≥ and k ≥ are integers. Then AQ n,k is (8 n − -conditional strongly Menger edge connected. Proof.
Let F be an arbitrary conditional faulty edge set of AQ n,k with | F | ≤ n −
10. Then δ ( AQ n,k − F ) ≥
2. Let u and v be any two distinct vertices in AQ n,k − F such that deg AQ n,k − F ( u ) ≤ deg AQ n,k − F ( v ).Also, let d u = deg AQ n,k − F ( u ) and let d v = deg AQ n,k − F ( v ). From Proposition 1, we will show that u and v are connected by d u edge-disjoint fault-free paths in AQ n,k − F . This means that u and v are stillconnected if the number of edges deleted is no more than d u − AQ n,k − F .Suppose, on the contrary, that u and v are separated by deleting a set of edges E f with | E f | ≤ d u − ≤ d v − AQ n,k − F . Let S = F ∪ E f . That is, u and v are disconnected in AQ n,k − S . Since d u = deg AQ n,k − F ( u ) ≤ deg AQ n,k ( u ) = 4 n −
2, we have | E f | ≤ n −
3. Then | S | ≤ (8 n −
10) + (4 n −
3) =12 n −
13. By Lemma 9, there is a component H in AQ n,k − S such that | V ( H ) | ≥ | V ( AQ n,k ) |−
2. It meansthat there are at most two vertices in AQ n,k − S not belonging to H . Clearly, | V ( H ) | (cid:54) = | V ( AQ n,k ) | , forotherwise, both u and v are contained in H .If | V ( H ) | = | V ( AQ n,k ) | −
1, without loss of generality, assume u is a singleton in AQ n,k − S . Clearly, E ( { u } , N AQ n,k − F ( u )) ⊆ E f . Thus, | E f | ≥ | N AQ n,k − F ( u ) | = deg AQ n,k − F ( u ) = d u , which contradict to | E f | ≤ d u − | V ( H ) | = | V ( AQ n,k ) | −
2. Let x and y be the two vertices which are not belonging to H in AQ n,k − S . Consider the following two cases:Case 1: x and y are adjacent in AQ n,k − S .Since u and v are separated in AQ n,k − S , without loss of generality, we assume that u ∈ V ( H )and v = x . Clearly, E ( { x, y } , N AQ n,k − F ( { x, y } )) ⊆ E f . Since F is a conditional faulty edge set, d AQ n,k − F ( y ) ≥
2, thus there is at least one edge except ( x, y ) which is incident with y in E f . Thus, | E f | ≥ | N AQ n,k − F ( { x, y } ) | ≥ (deg AQ n,k − F ( x ) −
1) + 1 = d v , which contradicts to | E f | ≤ d v − x and y are not adjacent in AQ n,k − S .Since u and v are separated in AQ n,k − S , we have { u, v } ∩ { x, y } (cid:54) = ∅ . Without loss of generality,assume u = x . Clearly, E ( { u } , N AQ n,k − F ( u )) ⊆ E f . Thus, | E f | ≥ | N AQ n,k − F ( u ) | = deg AQ n,k − F ( u ) = d u , which contradicts to | E f | ≤ d u − (cid:3) emark 4. To show that AQ n,k is not (8 n − C = ( u, u , u , u ) be a 3-cycle in AQ n,k . Also, let u ∈ N ( u ) \{ u, u } and v ∈ V ( AQ n,k ) \ ( N ( u ) ∪ N ( u )). Let F = (cid:83) i =1 E ( u i , N ( u i )) \ [ E ( C ) ∪ ( u , u )] bea faulty subset of edges in AQ n,k (i.e., edges with cross marks in Fig. 4). Clearly, | F | = 2(4 n − − − n − n − u and v in AQ n,k − F . Sincedeg AQ n,k − F ( u ) = deg AQ n,k − F ( v ) = 4 n − AQ n,k is not (8 n − N ( u )
In literature, there are many papers with results of computing strong Menger (edge) connectivity inseveral popular classes of triangle-free graphs. In this paper, we study the strong Menger (edge) con-nectivity of one kind of graph which has many triangles, namely augmented k -ary n -cube AQ n,k . Byexploring and utilizing the structural properties of AQ n,k , we show that AQ n, (resp. AQ n,k , k ≥
4) is(4 n − n − n ≥
3, and AQ n,k is (4 n − n ≥ k ≥
3. Moreover, under the restricted condition that each vertexhas at least two fault-free edges, we show that AQ n,k is (8 n − n ≥ k ≥
3. All results we obtained are optimal in the sense of the maximum numberof tolerated vertex (resp. edge) faults. Intuitively, we think that this method can also be applied to other r -regular r -connected graphs with triangles. References [1] H.-Y. Cai, H.-Q. Liu and M. Lu, Fault-tolerant maximal local-connectivity on Bubble-sort star graphs,Discrete Appl. Math. 181 (2015) 33–40.[2] Y.-C. Chen, M.-H. Chen and J.J.M. Tan, Maximally local connectivity and connected components of aug-mented cubes, Inform. Sci. 273 (2014) 387–392.[3] Q. Cheng, P. S. Li and M. Xu, Conditional (edge) fault-tolerant strong Menger (edge) connectivity of foldedhypercubes, Theoret. Comput. Sci. 728 (2018) 1–8.[4] S.A. Choudum and V. Sunitha, Augmented cubes, Networks 40(2) (2002) 71–84.[5] M.-M. Gu, R.-X. Hao, Y.-Q. Feng, The pessimistic diagnosability of bubble-sort star graphs and augmented k -ary n -cubes, Int. J. Comput. Math.: Comput. Sys. Theory 1 (2016) 98–112.
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Let S be an arbitrary set of edges in AQ ,k for k ≥ . If | S | ≤ , then there exists a component H in AQ ,k − S such that | V ( H ) | ≥ | V ( AQ ,k ) | − . Proof.
Let H be the large component of AQ ,k − S . By Lemma 1(2), the result holds if | S | ≤ κ ( AQ ,k ) − ≤ | S | ≤
9. Note that AQ ,k contains k disjoint copies of k -cycle, say AQ i ,k , i ∈ [ k ] .Let S i = S ∩ E ( AQ i ,k ) and s i = | S i | for i ∈ [ k ] . Let I = { i ∈ [ k ] : AQ i ,k − S i is disconnected } and J = [ k ] \ I . Clearly, s i ≥ i ∈ I . So | I | ≤ min { k, } . Let M be the union of smaller componentsof AQ n,k − S . We consider the following cases.Case 1. | I | = 0.For j ∈ [ k ] , AQ j ,k − S j is connected. By Lemma 1(3), there are 2 k edges between adjacent subgraphs AQ j ,k and AQ j +12 ,k for j ∈ [ k ] . Since k ≥ | S | ≤ < × (2 k ), there exists at most one integer,say i ∈ [ k ] , such that all the edges between AQ i ,k and AQ i +12 ,k are faulty. Since there is a fault-free edgebetween AQ j ,k − S j and AQ j +12 ,k − S j +1 for each j ∈ [ k ] \ { i } , it implies that AQ ,k − S is connected.Thus, | V ( H ) | = | V ( AQ ,k ) | .Case 2. | I | = 1. 16ithout loss of generality, assume that I = { } . So, V ( M ) ⊂ V ( AQ ,k ). By Lemma 1(2), s ≥ n − − j ∈ J , AQ j ,k − S j is connected. By Lemma 1(3), there are 2 k edges betweenadjacent subgraphs AQ j ,k and AQ j +12 ,k for j ∈ [ k ] . Since s c ≤ | S | − s ≤ − < k for k ≥ k ≥
4, there exists a fault-free edge between AQ j ,k − S j and AQ j +12 ,k − S j +1 for all j ∈ [ k ] . It leadsto AQ J ,k − S J is connected. Since every vertex in AQ ,k has four distinct extra neighbors, we have4 | V ( M ) | ≤ s c ≤
7. Thus, | V ( M ) | ≤ | V ( H ) | ≥ | V ( AQ ,k ) | − k = 3 (see Fig. 1(a)), then AQ , − S is either three singletons or an edge together with a singleton.Note that every vertex in AQ , has four distinct extra neighbors and 2 ≤ s ≤
3. Since s +4 | V ( M ) | ≤ | V ( M ) | ≤
1. Thus, at least a vertex of AQ , − S is connected to AQ , − S and AQ , − S .It implies | V ( H ) | ≥ | V ( AQ , ) | − | I | = 2.Without loss of generality, assume that I = { , t } where t ∈ [ k ] \ { } . For j ∈ J , AQ j ,k − S j isconnected. For i ∈ I , it is clear that s i ≥
2. Let H i be the large component of AQ i ,k − S i . Since s c ≤ | S | − s − s t ≤ − × AQ i ,k has four distinct extra neighbors, AQ i ,k − S i has two components, one is H i and the other is a singleton, say v i . By Lemma 1(3), there are 2 k edgesbetween adjacent subgraphs AQ j ,k and AQ j +12 ,k for j ∈ [ k ] . Since s c ≤ < k for k ≥
3, all H i ’s for i ∈ I and all subgraphs ( AQ j ,k − S j )’s for j ∈ J belong to the same component (i.e., H ) in AQ ,k − S .Since the two singletons, v in AQ ,k − S − H and v t in AQ t ,k − S t − H t , may be adjacent in AQ ,k − S ,we have s + s t + (4 | V ( M ) | − ≤
9. Thus, | V ( M ) | ≤ | V ( H ) | ≥ | V ( AQ ,k ) | − ≤ | I | ≤ min { k, } .In this case, k ≥
3. For j ∈ J , AQ j ,k − S j is connected. For i ∈ I , it is clear that s i ≥
2. Let H i be the large component of AQ i ,k − S i . By Lemma 1(3), there are 2 k edges between adjacent subgraphs AQ j ,k and AQ j +12 ,k for j ∈ [ k ] . Since s c ≤ | S | − | I | ≤ − × < k , all H i ’s for i ∈ I and allsubgraphs ( AQ j ,k − S j )’s for j ∈ J belong to the same component (i.e., H ) in AQ ,k − S . Since s c ≤ AQ i ,k has four distinct extra neighbors, if a vertex of AQ i ,k − S i is a singleton, thenit must be connected to H . Thus, | V ( H ) | ≥ | V ( AQ ,k ) | . (cid:3) Appendix B
Let S be an arbitrary set of edges in AQ ,k for k ≥ . If | S | ≤ , then there exists a component H in AQ ,k − S such that | V ( H ) | ≥ | V ( AQ ,k ) | − . Proof.
Let n = 2 and H be the large component of AQ ,k − S . By Lemma 8, the result holds if | S | ≤ n − ≤ | S | ≤
11. Recall that AQ ,k contains k disjoint copiesof k -cycle, say AQ i ,k , i ∈ [ k ] . Let S i = S ∩ E ( AQ i ,k ) and s i = | S i | for i ∈ [ k ] . Let I = { i ∈ [ k ] : AQ i ,k − S i is disconnected } and J = [ k ] \ I . Clearly, s i ≥ i ∈ I . So | I | ≤ min { k, } .Let M be the union of smaller components of AQ n,k − S . We consider the following cases.Case 1. | I | = 0.For j ∈ [ k ] , AQ j ,k − S j is connected. By Lemma 1(3), there are 2 k edges between adjacent subgraphs AQ j ,k and AQ j +12 ,k for j ∈ [ k ] . Since k ≥ | S | ≤ < × (2 k ), there exists at most one integer,say i ∈ [ k ] , such that all the edges between AQ i ,k and AQ i +12 ,k are faulty. Since there is a fault-free edgebetween AQ j ,k − S j and AQ j +12 ,k − S j +1 for each j ∈ [ k ] \ { i } , it implies that AQ ,k − S is connected.Thus, | V ( H ) | = | V ( AQ ,k ) | .Case 2. | I | = 1.Without loss of generality, assume that I = { } . So, V ( M ) ⊂ V ( AQ ,k ). By Lemma 1(2), s ≥ n − − j ∈ J , AQ j ,k − S j is connected. By Lemma 1(3), there are 2 k edges betweenadjacent subgraphs AQ j ,k and AQ j +12 ,k for j ∈ [ k ] . Since s c ≤ | S | − s ≤ − < k for k ≥ k ≥
5, there exists a fault-free edge between AQ j ,k − S j and AQ j +12 ,k − S j +1 for all j ∈ [ k ] . It leadsto AQ J ,k − S J is connected. Since every vertex in AQ ,k has four distinct extra neighbors, we have4 | V ( M ) | ≤ s c ≤
9. Thus, | V ( M ) | ≤ | V ( H ) | ≥ | V ( AQ ,k ) | − k = 3 (see Fig. 1(a)), then AQ , − S is either three singletons or an edge together with asingleton. Note that every vertex in AQ , has four distinct extra neighbors and 2 ≤ s ≤
3. Since s + 4 | V ( M ) | ≤
11, we have | V ( M ) | ≤
2. Thus, at least a vertex of AQ , − S is connected to AQ , − S and AQ , − S . It implies | V ( H ) | ≥ | V ( AQ , ) | − k = 4 (see Fig. 1(b)), then AQ , − S is one of the following: (1) four singletons; (2) an edge andtwo singletons; (3) a 2-path (i.e., a path of length 2) and a singleton; (4) two nonadjacent edges. Notethat every vertex in AQ , has four distinct extra neighbors and 2 ≤ s ≤
4. Since s + 4 | V ( M ) | ≤ | V ( M ) | ≤
2, and M is either an edge or at most two singletons. Thus, there exists either anedge, a 2-path, or two singletons of AQ , − S , which and all subgraphs ( AQ j , − S j )’s for j ∈ J belongto the same component (i.e., H ) in AQ ,k − S . This shows that | V ( H ) | ≥ | V ( AQ , ) | − | I | = 2.Without loss of generality, assume that I = { , t } where t ∈ [ k ] \ { } . For j ∈ J , AQ j ,k − S j is connected. For i ∈ I , it is clear that s i ≥
2. Let H i be the large component of AQ i ,k − S i . Since s c ≤ | S | − s − s t ≤ − × AQ i ,k has four distinct extra neighbors, AQ i ,k − S i has two components, one is H i and the other is a singleton, say v i . By Lemma 1(3), there are 2 k edgesbetween adjacent subgraphs AQ j ,k and AQ j +12 ,k for j ∈ [ k ] . Since s c ≤ < × (2 k ) for k ≥
3, all H i ’s for i ∈ I and all subgraphs ( AQ j ,k − S j )’s for j ∈ J belong to the same component (i.e., H ) in AQ ,k − S .Since the two singletons, v in AQ ,k − S − H and v t in AQ t ,k − S t − H t , may be adjacent in AQ ,k − S ,we have s + s t + (4 | V ( M ) | − ≤
11. Thus, | V ( M ) | ≤ | V ( H ) | ≥ | V ( AQ ,k ) | − | I | = 3.Without loss of generality, assume that I = { , t, p } where t, p ∈ [ k ] \ { } . For j ∈ J , AQ j ,k − S j is connected. For i ∈ I , it is clear that s i ≥
2. Let H i be the large component of AQ i ,k − S i . Since s c ≤ | S | − s − s t − s p ≤ − × AQ i ,k has four distinct extra neighbors, AQ i ,k − S i has two components, one is H i and the other is a singleton, say v i . By Lemma 1(3), there are2 k edges between adjacent subgraphs AQ j ,k and AQ j +12 ,k for j ∈ [ k ] . Since s c ≤ < × (2 k ) for k ≥ H i ’s for i ∈ I and all subgraphs ( AQ j ,k − S j )’s for j ∈ J belong to the same component (i.e., H ) in AQ ,k − S . If every two singletons of { v , v t , v p } are adjacent in AQ ,k − S , then s + s t + s p +(4 × − ≥
12, a contradiction. Thus at least two singletons in { v , v t , v p } are nonadjacent, and it follows that s + s t + s p + (4 | V ( M ) | − ≤
11. This shows that | V ( M ) | ≤ | V ( H ) | ≥ | V ( AQ , ) | − ≤ | I | ≤ min { k, } .In this case, k ≥
4. For j ∈ J , AQ j ,k − S j is connected. For i ∈ I , it is clear that s i ≥
2. Let H i be the large component of AQ i ,k − S i . By Lemma 1(3), there are 2 k edges between adjacent subgraphs AQ j ,k and AQ j +12 ,k for j ∈ [ k ] . Since s c ≤ | S | − | I | ≤ − × < k , all H i ’s for i ∈ I and allsubgraphs ( AQ j ,k − S j )’s for j ∈ J belong to the same component (i.e., H ) in AQ ,k − S . Since s c ≤ AQ i ,k has four distinct extra neighbors, if a vertex of AQ i ,k − S i is a singleton, thenit must be connected to H . Thus, | V ( H ) | ≥ | V ( AQ ,k ) | . (cid:3)(cid:3)