A Degree Condition for Graphs Having All(a,b)-parity Factors

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.

Degree Conditions for k-Hamiltonian[a,b]-factors

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.

Degree Conditions Restricted to Induced Paths for Hamiltonicity of Claw-heavy Graphs

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.

A New Sufficient Degree Condition for a Graphic Sequence to Be Forcibly k-Edge-Connected

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.

Degree sum conditions for hamiltonian index

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 Note on Z_3 -Connected Graphs with Degree Sum Condition

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.

Degree sum of a pair of independent edges and Z3-connectivity

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].

2-Factors with a Few Components in Balanced Bipartite Graphs

In this paper, a sufficient condition for a balanced bipartite graph to contain a 2-factor F is given. We show that every balanced bipartite graph of order 2n (n≥6)and e(G)>n2−2n+4contains a 2-factor with k components, 2d1-cycle, ⋯, 2dk-cycle, if one of the following is satisfied: (1) k=2, δ(G)≥2and d1−2≥d2≥2;(2) k=3, δ(G)≥d3+2and d1−2≥d2≥d3≥4. In particular, this extends one result of Moon and Moser in 1963 under condition (1).

A Vertex Cover with Chorded 4-cycles

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.

Disjoint Cliques in Claw-free Graphs

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.

Any Long Cycles Covering Specified Independent Vertices

An invariant σ2（G） of a graph is defined as follows： σ2（G） ：= min{d（u） ＋ d（v）｜u, v ∈V（G）,uv ∈ E（G）,u ≠ v} is the minimum degree sum of nonadjacent vertices （when G is a complete graph, we define σ2（G） = ∞）. Let k, s be integers with k ≥ 2 and s ≥ 4, G be a graph of order n sufficiently large compared with s and k. We show that if σ2（G） ≥ n ＋ k- 1, then for any set of k independent vertices v1,..., vk, G has k vertex-disjoint cycles C1,..., Ck such that ｜Ci｜ ≤ s and vi ∈ V（Ci） for all 1 ≤ i ≤ k. The condition of degree sum σs（G） ≥ n ＋ k - 1 is sharp.

SOLVABILITY FOR FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATION BOUNDARY VALUE PROBLEMS AT RESONANCE

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.