Let a,b and k be nonnegative integers with a≥2 and b≥a(k+1)+2.A graph G is called a k-Hamiltonian graph if after deleting any k vertices of G the remaining graph of G has a Hamiltonian cycle.A graph G is said to hav...Let a,b and k be nonnegative integers with a≥2 and b≥a(k+1)+2.A graph G is called a k-Hamiltonian graph if after deleting any k vertices of G the remaining graph of G has a Hamiltonian cycle.A graph G is said to have a k-Hamiltonian[a,b]-factor if after deleting any k vertices of G the remaining graph of G admits a Hamiltonian[a,b]-factor.Let G is a k-Hamiltonian graph of order n with n≥a+k+2.In this paper,it is proved that G contains a k-Hamiltonian[a,b]-factor ifδ(G)≥a+k andδ(G)≥I(G)≥a-1+(a(k+1))/(b-2).展开更多
Let a,b,k be nonnegative integers with 2≤a<b.A graph G is called a k-Hamiltonian graph if G-U contains a Hamiltonian cycle for any subset U?V(G)with|U|=k.An[a,b]-factor F of G is called a Hamiltonian[a,b]-factor i...Let a,b,k be nonnegative integers with 2≤a<b.A graph G is called a k-Hamiltonian graph if G-U contains a Hamiltonian cycle for any subset U?V(G)with|U|=k.An[a,b]-factor F of G is called a Hamiltonian[a,b]-factor if F contains a Hamiltonian cycle.If G-U admits a Hamiltonian[a,b]-factor for any subset U?V(G)with|U|=k,then we say that G has a k-Hamiltonian[a,b]-factor.Suppose that G is a k-Hamiltonian graph of order n with n≥((a+b-4)(2 a+b+k-6))/(b-2)+k andδ(G)≥a+k.In this paper,it is proved that G admits a k-Hamiltonian[a,b]-factor if max{dG(x),dG(y)}≥((a-2)n+(b-2)k)/(a+b-4)+2 for each pair of nonadjacent vertices x and y in G.展开更多
A digraph D is k-ordered if for every sequence S：v 1,v 2,…,v k of k distinct vertices,there exists a cycle C such that C encounters the vertices of S in the specified order.In particular,we say that D is k-ordered h...A digraph D is k-ordered if for every sequence S：v 1,v 2,…,v k of k distinct vertices,there exists a cycle C such that C encounters the vertices of S in the specified order.In particular,we say that D is k-ordered hamiltonian if for every sequence S：v 1,v 2,…,v k of k distinct vertices,there exists a hamiltonian cycle C such that the vertices of S are encountered on C in the specified order.In this paper,sufficient conditions for digraphs to be ordered and ordered hamiltonian have been given.展开更多
Projected Runge-Kutta （R-K） methods for constrained Hamiltonian systems are proposed. Dynamic equations of the systems, which are index-3 differential-algebraic equations （DAEs） in the Heisenberg form, are establi...Projected Runge-Kutta （R-K） methods for constrained Hamiltonian systems are proposed. Dynamic equations of the systems, which are index-3 differential-algebraic equations （DAEs） in the Heisenberg form, are established under the framework of Lagrangian multipliers. R-K methods combined with the technique of projections are then used to solve the DAEs. The basic idea of projections is to eliminate the constraint violations at the position, velocity, and acceleration levels, and to preserve the total energy of constrained Hamiltonian systems by correcting variables of the position, velocity, acceleration, and energy. Numerical results confirm the validity and show the high precision of the proposed method in preserving three levels of constraints and total energy compared with results reported in the literature.展开更多
A graph G has the hourglass property if every induced hourglass S（a tree with a degree sequence 22224） contains two non-adjacent vertices which have a common neighbor in G-V（S）.For an integer k≥4,a graph G has th...A graph G has the hourglass property if every induced hourglass S（a tree with a degree sequence 22224） contains two non-adjacent vertices which have a common neighbor in G-V（S）.For an integer k≥4,a graph G has the single k-cycle property if every edge of G,which does not lie in a triangle,lies in a cycle C of order at most k such that C has at least「｜V（C） /2」 edges which do not lie in a triangle,and they are not adjacent.In this paper,we show that every hourglass-free claw-free graph G of δ（G） ≥3 with the single 7-cycle property is Hamiltonian and is best possible;we also show that every claw-free graph G of δ（G） ≥3 with the hourglass property and with single 6-cycle property is Hamiltonian.展开更多
基金supported by the National Natural Science Foundation of China (Grant No. 11371009)the National Social Science Foundation of China (Grant No. 14AGL001)+1 种基金sponsored by Six Big Talent Peak of Jiangsu Province (Grant No. JY–022)333 Project of Jiangsu Province
文摘Let a,b and k be nonnegative integers with a≥2 and b≥a(k+1)+2.A graph G is called a k-Hamiltonian graph if after deleting any k vertices of G the remaining graph of G has a Hamiltonian cycle.A graph G is said to have a k-Hamiltonian[a,b]-factor if after deleting any k vertices of G the remaining graph of G admits a Hamiltonian[a,b]-factor.Let G is a k-Hamiltonian graph of order n with n≥a+k+2.In this paper,it is proved that G contains a k-Hamiltonian[a,b]-factor ifδ(G)≥a+k andδ(G)≥I(G)≥a-1+(a(k+1))/(b-2).
基金supported by the National Natural Science Foundation of China(Grant No.11371009)the National Social Science Foundation of China(Grant No.14AGL001)+1 种基金sponsored by Six Big Talent Peak of Jiangsu Province(Grant No.JY–022)333 Project of Jiangsu Province。
文摘Let a,b,k be nonnegative integers with 2≤a<b.A graph G is called a k-Hamiltonian graph if G-U contains a Hamiltonian cycle for any subset U?V(G)with|U|=k.An[a,b]-factor F of G is called a Hamiltonian[a,b]-factor if F contains a Hamiltonian cycle.If G-U admits a Hamiltonian[a,b]-factor for any subset U?V(G)with|U|=k,then we say that G has a k-Hamiltonian[a,b]-factor.Suppose that G is a k-Hamiltonian graph of order n with n≥((a+b-4)(2 a+b+k-6))/(b-2)+k andδ(G)≥a+k.In this paper,it is proved that G admits a k-Hamiltonian[a,b]-factor if max{dG(x),dG(y)}≥((a-2)n+(b-2)k)/(a+b-4)+2 for each pair of nonadjacent vertices x and y in G.
基金Foundation item： Supported by the National Natural Science Foundation of China（61070229） Supported by the Natural Science Foundation of Shanxi Province（2008011010）
文摘A digraph D is k-ordered if for every sequence S：v 1,v 2,…,v k of k distinct vertices,there exists a cycle C such that C encounters the vertices of S in the specified order.In particular,we say that D is k-ordered hamiltonian if for every sequence S：v 1,v 2,…,v k of k distinct vertices,there exists a hamiltonian cycle C such that the vertices of S are encountered on C in the specified order.In this paper,sufficient conditions for digraphs to be ordered and ordered hamiltonian have been given.
基金Project supported by the National Natural Science Foundation of China(No.11432010)the Doctoral Program Foundation of Education Ministry of China(No.20126102110023)+2 种基金the 111Project of China(No.B07050)the Fundamental Research Funds for the Central Universities(No.310201401JCQ01001)the Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University(No.CX201517)
文摘Projected Runge-Kutta （R-K） methods for constrained Hamiltonian systems are proposed. Dynamic equations of the systems, which are index-3 differential-algebraic equations （DAEs） in the Heisenberg form, are established under the framework of Lagrangian multipliers. R-K methods combined with the technique of projections are then used to solve the DAEs. The basic idea of projections is to eliminate the constraint violations at the position, velocity, and acceleration levels, and to preserve the total energy of constrained Hamiltonian systems by correcting variables of the position, velocity, acceleration, and energy. Numerical results confirm the validity and show the high precision of the proposed method in preserving three levels of constraints and total energy compared with results reported in the literature.
基金Supported by the National Natural Science Foundation of China(11071016 and 11171129)the Beijing Natural Science Foundation(1102015)
文摘A graph G has the hourglass property if every induced hourglass S（a tree with a degree sequence 22224） contains two non-adjacent vertices which have a common neighbor in G-V（S）.For an integer k≥4,a graph G has the single k-cycle property if every edge of G,which does not lie in a triangle,lies in a cycle C of order at most k such that C has at least「｜V（C） /2」 edges which do not lie in a triangle,and they are not adjacent.In this paper,we show that every hourglass-free claw-free graph G of δ（G） ≥3 with the single 7-cycle property is Hamiltonian and is best possible;we also show that every claw-free graph G of δ（G） ≥3 with the hourglass property and with single 6-cycle property is Hamiltonian.
