Fault Hamiltonian properties of the (n,k)-star graphs

Fault Hamiltonian properties of the (n,k)-star graphs Hong-Chun Hsu, Y.L. Hsieh, J.M. Tan, and L.H. Hsu Networks, Vol 42(4), pp. 189-201.

Star Graph, Sn • The vertex set of Sn is the permutation of n digits. • The adjacency is defined as follows: u1u2… ui-1uiui+1… un is adjacent to uiu2… ui-1u1ui+1… un, i.e. swap u1 and ui.

Example: S4 • The vertex set is {1234,2134,3124,1324,2314,3214, 4231,2431,3421,4321,2341,3241, 4132,1432,3412,4312,1342,3142, 4123,1423,2413,4213,1243,2143}

Example: S4 • Examples of edge: 1234 is connected to 1 2 1 3 4 1 2 3 4 1 2 3 4

Example: S4

(n,k)-star graph • The (n,k)-star graph was proposed by Chiang and Chen as a generalization of the star graph. • The (n,k)-star graph, Sn,k is a graph with the vertex set • V(Sn,k)= {u1 u2… uk | ui in <n> and ui uj for i  j }. • The permutation of k digits take from n digits.

(n,k)-star graph • The edge set: a vertex u1u2…ui…uk is adjacent to • (1) the vertex uiu2…u1…uk, where 2  i  k (i.e. swap ui with u1), and • (2) the vertex xu2u3… uk, where x in <n> -{ ui | 1  i  k}.

Example of (6,4)-star • The vertex 1234 is adjacent to: 1 2 3 4 1 2 3 4 1 2 3 4 1 5 2 3 4 1 6 2 3 4

Fault-tolerant Hamiltonian • In this paper, we consider the fault hamiltonicity and the fault hamiltonian connectivity of the (n,k)-star graph Sn,k.

Fault-tolerant Hamiltonian • We have the following results: • Let F be a faulty set of a graph G. • FV(G)E(G) • Sn,k-F is hamiltonian if |F|n-3 with n4 and n-k2. • Sn,k-F is hamiltonian connected if |F|n-4 with n4 and n-k2.

Fault-tolerant Hamiltonian

Some properties of complete graphs • Theorem 2[Ore]: Assume that G=(V,E) is a graph with n vertices with n > 3. Then G is hamiltonian if |Ē|n-3, and hamiltonian connected if |Ē|n-4.

Some properties of complete graphs • Theorem 3: Assume that G=(V,E) is a graph with V = < n >, n  4, and |Ē| n-4. Then, there are two hamiltonian paths of G joining any two different vertices i and j in V, say P1 = < i = i1, i2, …, in-1, in = j > and P2 = < i=i'1, i'2, …, i'n-1, i'n=j >, such that i2 i'2 and in-1 i'n-1.

Proof of Theorem 3 • Case 1: |{x<n-1>|(n,x)Ē}|=0 • Case 2: |{x<n-1>|(n,x)Ē}|1

Case 1: |{x<n-1>|(n,x)Ē}|=0 Suppose that i=n or j=n. <i,an-1,...,a2,j> is one hamiltonian path. Since | Ē<n-1>|=(n-1)-3, there is a hamiltonian cycle of G<n-1> an-1 i=n j a2 <i,a2,...,an-1,j> is the other hamiltonian path.

Case 1: |{x<n-1>|(n,x)Ē}|=0 Suppose that in and j n. Choose any edge e in Ē<n-1>. Hence, <i,...,a1t,n,a1t+1,...,j> is one hamiltonian path. i There are two hamiltonian paths in G<n-1>+{e}. a1t a2t n e e a2t+1 a1t+1 <i,...,a2t,n,a2t+1,...,j> is the other hamiltonian path. j

Case 2: |{x<n-1>|(n,x)Ē}|1 Suppose that i=n or j=n. We may assume that r and s are not j. Since | Ē|n-4, there are at least 3 vertices r, s, and t, such that {(r,n), (s,n), (t,n)}<n>. By induction, there are two hamiltonian paths of G<n-1> joining s and j. r s i=n Also, there are two hamiltonian paths of G<n-1> joining r and j. j Thus, there are two hamiltonian paths of G joining i and j.

Case 2: |{x<n-1>|(n,x)Ē}|1 Suppose that (i,n) or (j,n) is inĒ. Suppose that in and jn. There are two hamiltonian paths of G<n-1> joining i and j. There exists a smallest index p such that (ap,n) in E. i J Set J={a1,a2,...,ap-1}. ap-1 ap Since | Ē-J|(n-4)-(p-1), there are two hamiltonian paths of G<n>-J joining ap and j. n j Thus, there is a hamiltonian path of G joining i and j. Similar, there is another hamiltonian path of G joining i and j.

Case 2: |{x<n-1>|(n,x)Ē}|1 Suppose that in and jn. Suppose that (i,n) and (j,n) is in E. Assume that |{x<n>|(n,x) Ē}|1 There is a hamiltonian cycle of G<n-1>-{i}. i n Thus, there is a hamiltonian path of G joining i and j. k j Similar, there is another hamiltonian path of G joining i and j.

Case 2: |{x<n-1>|(n,x)Ē}|1 Suppose that in and jn. Suppose that (i,n) and (j,n) is in E. Assume that |{x<n>|(n,x) Ē}|2 There exists a k in <n>-{i,j} such that (n,k) is in E. i There is a hamiltonian path of G<n-1>-{i} joining k and j. n k Thus, there is a hamiltonian path of G joining i and j. j Similar, there is another hamiltonian path of G joining i and j.

Some properties of complete graphs • Lemma 7: Assume that n  4. Then Kn is (n-3)-fault hamiltonian and (n-4)-fault hamiltonian connected.

Some properties of complete graphs • Theorem 4 [Huang]: Let Kn=(V,E) be the complete graph with n vertices. Let F (VE) be a faulty set with |F| n-2. Then there exists a vertex set V'  V(Kn)-F with |V'|=n-|F| such that there exists a hamiltonian path of Kn-F joining every pair of vertices in V'.

Some properties of (n,k)-star graph • (n,k)-star is (n-1) regular with n!/(n-k)! vertices. • The (n,k)-star graph is node symmetric. • The (n, n-1)-star graph Sn,n-1 is isomorphic to the n-star graph Sn. • The (n, 1)-star graph Sn,1 is isomorphic to the complete graph Kn.

Some properties of (n,k)-star graph • Let Sn-1,k-1(i) denote a subgraph of Sn,k induced by all the nodes with the (u)n= i, for some 1  i  n. • Sn,k can be decomposed into n subgraph Sn-1,k-1(i), 1  i  n, and • each subgraph Sn-1,k-1(i) is isomorphic to Sn-1,k-1.

Some properties of (n,k)-star graph • Ei,j to denote the set of edges between Sn-1,k-1(i) and Sn-1,k-1(j). • (u, v) be any edge in Ei,j. Then u Sn-1,k-1(i) and v Sn-1,k-1(j). • | Ei,j|= (n - 2)!/(n-k)!

Some properties of (n,k)-star graph • Lemma 8: Let n > k >1 and u and v be two distinct vertices in Sn-1,k-1(l) with d(u, v)  2 for some 1  l  n. Then (u)1 (v)1. • Lemma 9: Let (u, v) and (u', v') be any two distinct edges in Ei,j. Then, {u, v}  {u', v'} = .

Some properties of (n,k)-star graph • Let F be a faulty set of Sn,k. • An edge (u, v) is F-fault if (u, v)  F, u  F, or v  F; and • (u, v) is F-fault free if (u, v) is not F-fault. • Let H = (V', E') be a subgraph of Sn,k. We use F(H) to denote the set (V'  E')  F.

Some properties of (n,k)-star graph • We associate Sn,k with the complete graph Kn with vertex set <n> such that vertex l of Kn is associated with Sn-1, k-1(l) for every 1 l  n. • We define a faulty edge set R(F) of Kn as (i,j)  R(F) if some edge of Ei,j is F-fault. • We use Kn(I) to denote the subgraph of Kn induced by I.

Some properties of (n,k)-star graph • Lemma 10: Suppose that k 2, and (n-k) 2. Let I <n> with |I|=m  2 and let F  V(Sn,k) E(Sn,k) with Sn-1,k-1(i)-F being hamiltonian connected for all i I. Let u and v be any two vertices of Sn-1,k-1(l) such that • (1) (u)k (v)k, • (2) there exists a hamiltonian path P=< (u)k=i1,i2,… ,im=(v)k> of Kn(l)-R(F), and • (3) (u)1i2 and (v)1  im-1 if k=2. • Then there exists a hamiltonian path of Sn-1,k-1(I)-F joining u to v.

Hamiltonian properties of (n,k)-star graph • Lemma 11: S4,2 is 1-fault hamiltonian and hamiltonian connected. • Lemma 12: Suppose that, for some k  2, n  5, and n-k  2, Sn-1,k-1 is (n-4)-fault hamiltonian and (n-5)-fault hamiltonian connected. Then Sn,k is (n-3)-fault hamiltonian.

Fault Hamiltonian connected of (n,k)-star graph—Base case • Lemma 13: Sn,2 is (n-4)-fault hamiltonian connected for n 5. • Proof:

Fault Hamiltonian connected of (n,k)-star graph —Base case • Case 1: |F(Sn-1,1(1))|  n-5. • Subcase 1.1: (x)k= (y)k. • Subcase 1.2: (x)k (y)k. • Case 2: |F(Sn-1,1(1))| = n-4. • Subcase 2.1: (x)k= (y)k=1. • Subcase 2.2: (x)k= 1 and (y)k 1. • Subcase 2.3: (x)k= (y)k 1. • Subcase 2.4: (x)k , (y)k and 1 are distinct.

Fault Hamiltonian connected of (n,k)-star graph • Lemma 14: Suppose that, for some k  3, n  5, and n-k  2, Sn-1,k-1 is (n-4)-fault hamiltonian and (n-5)-fault hamiltonian connected. Then Sn,k is (n-4)-fault hamiltonian connected. • Proof.

Fault Hamiltonian connected of (n,k)-star graph • Case 1: |F(Sn-1,k-1(1))|  n-5. • Subcase 1.1: (x)k (y)k. • Subcase 1.2: (x)k= (y)k. • Case 2: |F(Sn-1,k-1(1))| = n-4. • Subcase 2.1: (x)k= (y)k=1. • Subcase 2.2: (x)k= 1 and (y)k 1. • Subcase 2.3: (x)k= (y)k 1. • Subcase 2.4: (x)k , (y)k and 1 are distinct

Subcase 1.1: (x)k (y)k. • By Lemma 10, there exists a hamiltonian path of Sn,k-F joining x and y.

Subcase 1.2: (x)k= (y)k There exists a hamiltonian path of S(x)kn-1,k-1-F joining x and y. There exists an edge (u,v). x y By Lemma 10, there exists a hamiltonian path joining x and y. v u

Subcase 2.1: (x)k= (y)k=1 There exists a hamiltonian path of S1n-1,k-1-F-{f} joining x and y. x y By Lemma 10, there exists a hamiltonian path joining x and y. u v f

Subcase 2.2: (x)k= 1 and (y)k 1 There exists a hamiltonian cycle of S1n-1,k-1-F. Assume that w is a neighbor of x such that (k(w))k(y)k w x By Lemma 10, there exists a hamiltonian path joining x and y. k(w) y

Subcase 2.3: (x)k= (y)k 1 There exists a hamiltonian path of Skn-1,k-1-F joining x and y. There exists a hamiltonian cycle in S1n-1,k-1-F. x y k(u) There exists an edge (u,w) such that (k(u))k=1. u w By Lemma 10, there exists a hamiltonian path joining x and y.

Subcase 2.4: (x)k , (y)k and 1 are distinct There exists a hamiltonian path of Skn-1,k-1-F joining x and u such that k(u) in S1. There exists a hamiltonian cycle in S1n-1,k-1-F. x k(u) v u By Lemma 10, there exists a hamiltonian path joining x and y. y k(v)

Hamiltonian properties of (n,k)-star graph • Theorem 5 Let n and k be two positive integers with n > k  1. Then • (1) Sn,k is (n-3)-fault hamiltonian and (n-4)-fault hamiltonian connected; • (2) S2,1 is hamiltonian connected; and • (3) Sn,n-1 is hamiltonian.

Fault Hamiltonian properties of the (n,k)-star graphs