Hua-Min Huang

The super connectivity of the pancake graphs and the super laceability of the star graphs

A k-container C(u, v) of a graph G is a set of k-disjoint paths joining u to v. A k-container C(u, v) of G is a k*-container if it contains all the vertices of G. A graph G is k*-connected if there exists a k*-container between any two distinct vertices. Let κ(G) be the connectivity of G. A graph G is super connected if G is i*-connected for all 1 ≤ i ≤ κ(G). A bipartite graph G is k*-laceable if there exists a k*-container between any two vertices from different parts of G. A bipartite graph G is super laceable if G is i*-laceable for all 1 ≤ i ≤ κ(G). In this paper, we prove that the n-dimensional pancake graph Pn is super connected if and only if n ≠ 3 and the n-dimensional star graph Sn is super laceable if and only if n ≠ 3.

The super laceability of the hypercubes

A k-container C(u,v) of a graph G is a set of k disjoint paths joining u to v. A k-container C(u, v) is a k*-container if every vertex of G is incident with a path in C(u,v). A bipartite graph G is k*-laceable if there exists a k*-container between any two vertices u, v from different partite set of G. A bipartite graph G with connectivity k is super laceable if it is i*-laceable for all i ≤ k. A bipartite graph G with connectivity k is f-edge fault-tolerant super laceable if G - F is i*-laceable for any 1 ≤ i ≤ k-f and for any edge subset F with |F|=f r is super laceable. Moreover, Q r is f-edge fault-tolerant super laceable for any f ≤ r - 2.

MUTUALLY INDEPENDENT HAMILTONIAN PATHS AND CYCLES IN HYPERCUBES

Two hamiltonian paths P1 = 〈v1, v2, …, vn(G) 〉 and P2 = 〈 u1, u2, …, un(G) 〉 of G are independent if v1 = u1, vn(G) = un(G), and vi ≠ ui for 1 < i < n(G). A set of hamiltonian paths {P1, P2, …, Pk} of G are mutually independent if any two different hamiltonian paths in the set are independent. A bipartite graph G is hamiltonian laceable if there exists a hamiltonian path joining any two nodes from different partite sets. A bipartite graph is k-mutually independent hamiltonian laceable if there exist k-mutually independent hamiltonian paths between any two nodes from different partite sets. The mutually independent hamiltonian laceability of bipartite graph G, IHPL(G), is the maximum integer k such that G is k-mutually independent hamiltonian laceable. Let Qn be the n-dimensional hypercube. We prove that IHPL(Qn) = 1 if n ∈ {1,2,3}, and IHPL(Qn) = n - 1 if n ≥ 4. A hamiltonian cycle C of G is described as 〈 u1, u2, …, un(G), u1 〉 to emphasize the order of nodes in C. Thus, u1 is the beginning node and ui is the i-th node in C. Two hamiltonian cycles of G beginning at u, C1 = 〈 v1, v2, …, vn(G), v1 〉 and C2 = 〈 u1, u2, …, un(G), u1 〉, are independent if u = v1 = u1, and vi ≠ ui for 1 < i ≤ n(G). A set of hamiltonian cycles {C1, C2, …, Ck} of G are mutually independent if any two different hamiltonian cycles are independent. The mutually independent hamiltonianicity of graph G, IHC(G), is the maximum integer k such that for any node u of G there exist k-mutually independent hamiltonian cycles of G starting at u. We prove that IHC(Qn) = n - 1 if n ∈ {1,2,3} and IHC(Qn) = n if n ≥ 4.

Note: On spanning connected graphs

A k-containerC(u,v) of G between u and v is a set of k internally disjoint paths between u and v. A k-container C(u,v) of G is a k^*-container if the set of the vertices of all the paths in C(u,v) contains all the vertices of G. A graph G is k^*-connected if there exists a k^*-container between any two distinct vertices. Therefore, a graph is 1^*-connected (respectively, 2^*-connected) if and only if it is hamiltonian connected (respectively, hamiltonian). In this paper, a classical theorem of Ore, providing sufficient conditional for a graph to be hamiltonian (respectively, hamiltonian connected), is generalized to k^*-connected graphs.

On the spanning connectivity of graphs

Abstract A k-container C ( u , v ) of G between u and v is a set of k internally disjoint paths between u and v . A k-container C ( u , v ) of G is a k * -container if it contains all vertices of G. A graph G is k * -connected if there exists a k * -container between any two distinct vertices. The spanning connectivity of G, κ * ( G ) , is defined to be the largest integer k such that G is w * -connected for all 1 ⩽ w ⩽ k if G is a 1 * -connected graph. In this paper, we prove that κ * ( G ) ⩾ 2 δ ( G ) - n ( G ) + 2 if ( n ( G ) / 2 ) + 1 ⩽ δ ( G ) ⩽ n ( G ) - 2 . Furthermore, we prove that κ * ( G - T ) ⩾ 2 δ ( G ) - n ( G ) + 2 - | T | if T is a vertex subset with | T | ⩽ 2 δ ( G ) - n ( G ) - 1 .

HAMILTONIAN LACEABILITY OF FAULTY HYPERCUBES

We consider the occurrence of the combination of edge faults and vertex faults in hypercubes. To preserve the equitability of Qn, we restrict the faults on vertex occurring only on disjoint adjacen...

The super spanning connectivity and super spanning laceability of the enhanced hypercubes

A k-containerC(u,v) of a graph G is a set of k disjoint paths between u and v. A k-container C(u,v) of G is a k*-container if it contains all vertices of G. A graph G is k*-connected if there exists a k*-container between any two distinct vertices of G. Therefore, a graph is 1*-connected (respectively, 2*-connected) if and only if it is Hamiltonian connected (respectively, Hamiltonian). A graph G is super spanning connected if there exists a k*-container between any two distinct vertices of G for every k with 1≤k≤κ(G) where κ(G) is the connectivity of G. A bipartite graph G is k*-laceable if there exists a k*-container between any two vertices from different partite set of G. A bipartite graph G is super spanning laceable if there exists a k*-container between any two vertices from different partite set of G for every k with 1≤k≤κ(G). In this paper, we prove that the enhanced hypercube Qn,m is super spanning laceable if m is an odd integer and super spanning connected if otherwise.

Mutually independent Hamiltonian cycles in hypercubes

A Hamiltonian cycle C of G is described as <u/sub 1/, u/sub 2/, ..., u/sub n(G)/, u/sub 1/> to emphasize the order of nodes in C. Thus, u/sub 1/ is the beginning node and u/sub i/ is the i-th node in C. Two Hamiltonian cycles of G beginning at u, C/sub 1/=<v/sub 1/, v/sub 2/, ..., v/sub n(G)/, v/sub 1/> and C/sub 2/=<u/sub 1/, u/sub 2/, ..., u/sub n(G)/, u/sub 1/>, are independent if u=v/sub 1/=u/sub 1/, and v/sub i//spl ne/u/sub i/ for 1<i/spl les/n(G). A set of Hamiltonian cycles {C/sub 1/, C/sub 2/, ..., C/sub k/} of G are mutually independent if any two different Hamiltonian cycles are independent. The mutually independent Hamiltonianicity of graph G, IHC(G), is the maximum integer k such that for any node u of G there exist k-mutually independent Hamiltonian cycles of G starting at u. Let Q/sub n/ be the n-dimensional hypercube. We prove that IHC(Q/sub n/)=n-1 if n/spl isin/{1, 2, 3} and IHC(Q/sub n/)=n if n/spl ges/4.

The spanning diameter of the star graphs

Assume that u and v are any two distinct vertices of different partite sets of S/sub n/ with n /spl ges/ 5. We prove that there are (n - 1) internally disjoint paths P/sub 1/, P/sub 2/, ..., P/sub n-i/ joining u to v such that /spl cup//sup n = 1//sub i = 2/ P/sub i/ spans S/sub n/ and l(P/sub i/) /spl les/ (n - 1)! + 2(n - 2)! + 2(n - 3)! + 1 = n!/(n - 2) + 1. We also prove that there are two internally disjoint paths Q/sub 1/ and Q/sub 2/ joining u to v such that Q/sub 1/ /spl cup/ Q/sub 2/ spans S/sub n/ and l(Q/sub i/) /spl les/ n!/2 + l for i = 1,2.

Mutually Independent Hamiltonianicity of Pancake Graphs and Star Graphs

A hamiltonian cycle C of a graph G is described as langu₁, u₂,..., u_n(G), u₁rang to emphasize the order of vertices in C. Thus, u₁ is the start vertex and u_i is the i-th vertex in C. Two hamiltonian cycles of G start at a vertex x, C₁ = langu₁, u₂,..., u_n(G), u₁rang and C₂ = langv₁, v₂,..., v_n(G), v₁rang, are independent if x = u₁ = v₁ and u₁ ne v_i for every i, 2 les i les n(G). A set of hamiltonian cycles {C₁, C₂,..., C_k} of G are mutually independent if any two different hamiltonian cycles are independent. The mutually independent hamiltonicity of graph G, IHC(G), is the maximum integer k such that for any vertex u of G there exist k-mutually independent hamiltonian cycles ofG starting at u. Inthispaper, we are going to study IHC(G) for the n-dimensional pancake graph P_n and the n-dimensional star graph S_n. We prove that IHC(P_n) = n - 1 if n ges 4 and IHC(S_n) = n-1 if nges5.