Let a and b be positive integers such that a≤b and a≡b(mod 2).We say that G has all(a,b)-parity factors if G has an h-factor for every function h:V(G)→{a,a+2,…,b-2,b} with b|V(G)| even and h(v)≡b(mod 2) for all v...Let a and b be positive integers such that a≤b and a≡b(mod 2).We say that G has all(a,b)-parity factors if G has an h-factor for every function h:V(G)→{a,a+2,…,b-2,b} with b|V(G)| even and h(v)≡b(mod 2) for all v∈V(G).In this paper,we prove that every graph G with n≥2(b+1)(a+b) vertices has all(a,b)-parity factors if δ(G)≥(b^(2)-b)/a,and for any two nonadjacent vertices u,v ∈V(G),max{d_(G)(u),d_(G)(v)}≥bn/a+b.Moreover,we show that this result is best possible in some sense.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
Broersma and Veldman proved that every 2-connected claw-free and P6-free graph is hamil- tonian. Chen et al. extended this result by proving every 2-connected claw-heavy and P6-free graph is hamiltonian. On the other ...Broersma and Veldman proved that every 2-connected claw-free and P6-free graph is hamil- tonian. Chen et al. extended this result by proving every 2-connected claw-heavy and P6-free graph is hamiltonian. On the other hand, Li et al. constructed a class of 2-connected graphs which are claw-heavy and P6-o-heavy but not hamiltonian. In this paper, we further give some Ore-type degree conditions restricting to induced copies of P6 of a 2-connected claw-heavy graph that can guarantee the graph to be hamiltonian. This improves some previous related results.展开更多
A graphic sequence π =(d1, d2,..., dn) is said to be forcibly k-edge-connected if every realization of π is k-edge-connected. In this paper, we obtain a new sufficient degree condition for π to be forcibly k-edgeco...A graphic sequence π =(d1, d2,..., dn) is said to be forcibly k-edge-connected if every realization of π is k-edge-connected. In this paper, we obtain a new sufficient degree condition for π to be forcibly k-edgeconnected. We also show that this new sufficient degree condition implies a strongest monotone degree condition for π to be forcibly 2-edge-connected and a conjecture about a strongest monotone degree condition for π to be forcibly 3-edge-connected due to Bauer et al.(Networks, 54(2)(2009) 95-98), and also implies a strongest monotone degree condition for π to be forcibly 4-edge-connected.展开更多
In this note,we show a sharp lower bound of min{Σ_(i=1)^(k)dG(u_(i)):u1u2...uk is a path of(2-)connected G}on its order such that(k-1)-iterated line graphs L^(k-1)(G)are hamiltonian.
A graph G satisfies the Ore-condition if d（x）＋d（y） ≥ ｜V（G）｜ for any xy E（G）. Luo et al. [European J. Combin., 2008] characterized the simple Z3-connected graphs satisfying the Ore-condition. In this paper...A graph G satisfies the Ore-condition if d（x）＋d（y） ≥ ｜V（G）｜ for any xy E（G）. Luo et al. [European J. Combin., 2008] characterized the simple Z3-connected graphs satisfying the Ore-condition. In this paper, we characterize the simple Z3-connected graphs G satisfying d（x）＋d（y） ≥ ｜V（G）｜ - 1 for any xy E（G）, which improves the results of Luo et al.展开更多
Let G be a 2-edge-connected simple graph on n vertices. For an edge e = uv ∈ E（G）, define d（e） = d（u） ＋ d（v）. Let F denote the set of all simple 2-edge-connected graphs on n ） 4 vertices such that G∈ F if ...Let G be a 2-edge-connected simple graph on n vertices. For an edge e = uv ∈ E（G）, define d（e） = d（u） ＋ d（v）. Let F denote the set of all simple 2-edge-connected graphs on n ） 4 vertices such that G∈ F if and only if d（e） ＋ d（e′） ≥ 2n for every pair of independent edges e, e′ of G. We prove in this paper that for each G ∈ F, G is not Z3-connected if and only if G is one of K2,n-2, K3,n-3, K^＋2,n-2,K^＋ 3,n-3 or one of the 16 specified graphs, which generalizes the results of X. Zhang et al. [Discrete Math., 20]0, 310： 3390-3397] and G. Fan and X. Zhou [Discrete Math., 2008, 308： 6233-6240].展开更多
Let k be an integer with k ≥ 2 and G a graph with order n 〉 4k. We prove that if the minimum degree sum of any two nonadjacent vertices is at least n ＋ k, then G contains a vertex cover with exactly k components su...Let k be an integer with k ≥ 2 and G a graph with order n 〉 4k. We prove that if the minimum degree sum of any two nonadjacent vertices is at least n ＋ k, then G contains a vertex cover with exactly k components such that k - 1 of them are chorded 4-cycles. The degree condition is sharp in general.展开更多
A graph is said to be claw-free if it does not contain an induced subgraph isomorphic to K_（1,3）. Let s and k be two integers with 0≤s≤k and let G be a claw-free graph of order n. In this paper, we investigate cli...A graph is said to be claw-free if it does not contain an induced subgraph isomorphic to K_（1,3）. Let s and k be two integers with 0≤s≤k and let G be a claw-free graph of order n. In this paper, we investigate clique partition problems in claw-free graphs. It is proved that if n≥3 s ＋4（k-s） and d（x）＋ d（y）≥n-2 s ＋2 k ＋1 for any pair of non-adjacent vertices x, y of G, then G contains s disjoint K3 s and k-s disjoint K4 s such that all of them are disjoint. Moreover, the degree condition is sharp in some cases.展开更多
The paper deals a fractional functional boundary value problems with integral boundary conditions. Besed on the coincidence degree theory, some existence criteria of solutions at resonance are established.
基金supported by the National Natural Science Foundation of China (No.12271425 and No.11871391)。
文摘Let a and b be positive integers such that a≤b and a≡b(mod 2).We say that G has all(a,b)-parity factors if G has an h-factor for every function h:V(G)→{a,a+2,…,b-2,b} with b|V(G)| even and h(v)≡b(mod 2) for all v∈V(G).In this paper,we prove that every graph G with n≥2(b+1)(a+b) vertices has all(a,b)-parity factors if δ(G)≥(b^(2)-b)/a,and for any two nonadjacent vertices u,v ∈V(G),max{d_(G)(u),d_(G)(v)}≥bn/a+b.Moreover,we show that this result is best possible in some sense.
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Supported by NSFC(Grant No.11271300)the Natural Science Foundation of Shaanxi Province(Grant No.2016JQ1002)the Project NEXLIZ–CZ.1.07/2.3.00/30.0038
文摘Broersma and Veldman proved that every 2-connected claw-free and P6-free graph is hamil- tonian. Chen et al. extended this result by proving every 2-connected claw-heavy and P6-free graph is hamiltonian. On the other hand, Li et al. constructed a class of 2-connected graphs which are claw-heavy and P6-o-heavy but not hamiltonian. In this paper, we further give some Ore-type degree conditions restricting to induced copies of P6 of a 2-connected claw-heavy graph that can guarantee the graph to be hamiltonian. This improves some previous related results.
基金supported by the Hainan Provincial Natural Science Foundation of China(No.2019RC085)the National Natural Science Foundation of China(No.11961019)。
文摘A graphic sequence π =(d1, d2,..., dn) is said to be forcibly k-edge-connected if every realization of π is k-edge-connected. In this paper, we obtain a new sufficient degree condition for π to be forcibly k-edgeconnected. We also show that this new sufficient degree condition implies a strongest monotone degree condition for π to be forcibly 2-edge-connected and a conjecture about a strongest monotone degree condition for π to be forcibly 3-edge-connected due to Bauer et al.(Networks, 54(2)(2009) 95-98), and also implies a strongest monotone degree condition for π to be forcibly 4-edge-connected.
基金Supported by the National Natural Science Foundation of China(11871099).
文摘In this note,we show a sharp lower bound of min{Σ_(i=1)^(k)dG(u_(i)):u1u2...uk is a path of(2-)connected G}on its order such that(k-1)-iterated line graphs L^(k-1)(G)are hamiltonian.
基金Supported by National Natural Science Foundation of China (Grant No. 11071233)the Fundamental Research Funds for the Central Universities (Grant No. WK0010000021)
文摘A graph G satisfies the Ore-condition if d（x）＋d（y） ≥ ｜V（G）｜ for any xy E（G）. Luo et al. [European J. Combin., 2008] characterized the simple Z3-connected graphs satisfying the Ore-condition. In this paper, we characterize the simple Z3-connected graphs G satisfying d（x）＋d（y） ≥ ｜V（G）｜ - 1 for any xy E（G）, which improves the results of Luo et al.
基金Acknowledgements The first author was supported by the Excellent Doctorial Dissertation Cultivation Grant from Huazhong Normal University （2013YBYB42）. The second author was supported in part by the National Natural Science Foundation of China （Grant No. 11171129） and the Doctoral Fund of Ministry of Education of China （Grant No. 20130144110001）.
文摘Let G be a 2-edge-connected simple graph on n vertices. For an edge e = uv ∈ E（G）, define d（e） = d（u） ＋ d（v）. Let F denote the set of all simple 2-edge-connected graphs on n ） 4 vertices such that G∈ F if and only if d（e） ＋ d（e′） ≥ 2n for every pair of independent edges e, e′ of G. We prove in this paper that for each G ∈ F, G is not Z3-connected if and only if G is one of K2,n-2, K3,n-3, K^＋2,n-2,K^＋ 3,n-3 or one of the 16 specified graphs, which generalizes the results of X. Zhang et al. [Discrete Math., 20]0, 310： 3390-3397] and G. Fan and X. Zhou [Discrete Math., 2008, 308： 6233-6240].
基金Supported by Natural Science Foundation of China （Grant Nos. 11161035, 10801091）, Research Crants from Ningxia University （Grant No. （E）ndzr09-1） and Scientific Research Project in Xinjiang （Grant No. XJEDU2009S101）
文摘Let k be an integer with k ≥ 2 and G a graph with order n 〉 4k. We prove that if the minimum degree sum of any two nonadjacent vertices is at least n ＋ k, then G contains a vertex cover with exactly k components such that k - 1 of them are chorded 4-cycles. The degree condition is sharp in general.
基金Supported by the National Natural Science Foundation of China(Grant No.11271230,11671232)
文摘A graph is said to be claw-free if it does not contain an induced subgraph isomorphic to K_（1,3）. Let s and k be two integers with 0≤s≤k and let G be a claw-free graph of order n. In this paper, we investigate clique partition problems in claw-free graphs. It is proved that if n≥3 s ＋4（k-s） and d（x）＋ d（y）≥n-2 s ＋2 k ＋1 for any pair of non-adjacent vertices x, y of G, then G contains s disjoint K3 s and k-s disjoint K4 s such that all of them are disjoint. Moreover, the degree condition is sharp in some cases.
基金Supported by the Fundamental Research Funds for the Central Universities
文摘The paper deals a fractional functional boundary value problems with integral boundary conditions. Besed on the coincidence degree theory, some existence criteria of solutions at resonance are established.
