5连通图中的可去边

An edge e of a k-connected graph G is said to be a removable edge if G O e is still k-connected, where G e denotes the graph obtained from G by deleting e to get G - e, and for any end vertex of e with degree k - 1 in G- e, say x, delete x, and then add edges between any pair of non-adjacent vertices in NG-e （x）. The existence of removable edges of k-connected graphs and some properties of 3-connected and 4-connected graphs have been investigated [1, 11, 14, 15]. In the present paper, we investigate some properties of 5-connected graphs and study the distribution of removable edges on a cycle and a spanning tree in a 5- connected graph. Based on the properties, we proved that for a 5-connected graph G of order at least 10, if the edge-vertex-atom of G contains at least three vertices, then G has at least （3│G│ ＋ 2）/2 removable edges.     

A conjecture on k-factor-critical and 3-γ-critical graphs

For a graph G =(V,E),a subset VS is a dominating set if every vertex in V is either in S or is adjacent to a vertex in S.The domination number γ(G) of G is the minimum order of a dominating set in G.A graph G is said to be domination vertex critical,if γ(G-v) γ(G) for any vertex v in G.A graph G is domination edge critical,if γ(G ∪ e) γ(G) for any edge e ∈/E(G).We call a graph G k-γ-vertex-critical(resp.k-γ-edge-critical) if it is domination vertex critical(resp.domination edge critical) and γ(G) = k.Ananchuen and Plummer posed the conjecture:Let G be a k-connected graph with the minimum degree at least k+1,where k 2 and k≡|V|(mod 2).If G is 3-γ-edge-critical and claw-free,then G is k-factor-critical.In this paper we present a proof to this conjecture,and we also discuss the properties such as connectivity and bicriticality in 3-γ-vertex-critical claw-free graph.     

Deleting Vertices and Interlacing Laplacian Eigenvalues

The authors obtain an interlacing relation between the Laplacian spectra of a graph G and its subgraph G - U, which is obtained from G by deleting all the vertices in the vertex subset U together with their incident edges. Also, some applications of this interlacing property are explored and this interlacing property is extended to the edge weighted graphs.     

<Emphasis Type="Italic">k</Emphasis>-factors in regular graphs

Plesnik in 1972 proved that an （m - 1）-edge connected m-regular graph of even order has a 1-factor containing any given edge and has another 1-factor excluding any given m - 1 edges. Alder et al. in 1999 showed that if G is a regular （2n ＋ 1）-edge-connected bipartite graph, then G has a 1-factor containing any given edge and excluding any given matching of size n. In this paper we obtain some sufficient conditions related to the edge-connectivity for an n-regular graph to have a k-factor containing a set of edges and （or） excluding a set of edges, where 1 ≤ k ≤n/2. In particular, we generalize Plesnik＇s result and the results obtained by Liu et al. in 1998, and improve Katerinis＇ result obtained 1993. Furthermore, we show that the results in this paper are the best possible.     

A Note on Strong Edge Coloring of Sparse Graphs

A strong edge coloring of a graph is a proper edge coloring where the edges at distance at most 2 receive distinct colors. The strong chromatic index χ'_s(G) of a graph G is the minimum number of colors used in a strong edge coloring of G. In an ordering Q of the vertices of G, the back degree of a vertex x of G in Q is the number of vertices adjacent to x, each of which has smaller index than x in Q. Let G be a graph of maximum degree Δ and maximum average degree at most 2 k. Yang and Zhu [J. Graph Theory, 83, 334–339(2016)] presented an algorithm that produces an ordering of the edges of G in which each edge has back degree at most 4 kΔ-2 k in the square of the line graph of G, implying that χ'_s(G) ≤ 4 kΔ-2 k + 1. In this note, we improve the algorithm of Yang and Zhu by introducing a new procedure dealing with local structures. Our algorithm generates an ordering of the edges of G in which each edge has back degree at most(4 k-1)Δ-2 k in the square of the line graph of G, implying that χ'_s(G) ≤(4 k-1)Δ-2 k + 1.     

3,4跳图的平面性

For a graph G of size ε≥1 and its edge-induced subgraphs H1 and H2 of size r(1≤r≤ε), H1 is said to be obtained from H2 by an edge jump if there exist four distinct vertices u,v,w and x in G such that (u,v)∈E(H2), (w,x)∈ E(G)-E(H2) and H1=H2-(u,v)+(w,x). In this article, the r-jump graphs (r≥3) are discussed. A graph H is said to be an r-jump graph of G if its vertices correspond to the edge induced graph of size r in G and two vertices are adjacent if and only if one of the two corresponding subgraphs can be obtained from the other by an edge jump. For k≥2, the k-th iterated r-jump graph Jrk(G) is defined as Jr(Jrk-1(G)), where Jr1(G)=Jr(G).An infinite sequence{Gi} of graphs is planar if every graph Gi is planar. It is shown that there does not exist a graph G for which the sequence {J3k(G)} is planar, where k is any positive integer. Meanwhile,lim gen(J3k(G))=∞,where gen(G) denotes the genus of a graph G, if the sequencek→∞J3k(G) is defined for every positive integer k. As for the 4-jump gra     

The Rainbow Vertex-disconnection in Graphs

Let G be a nontrivial connected and vertex-colored graph. A subset X of the vertex set of G is called rainbow if any two vertices in X have distinct colors. The graph G is called rainbow vertex-disconnected if for any two vertices x and y of G, there exists a vertex subset S of G such that when x and y are nonadjacent, S is rainbow and x and y belong to different components of G-S; whereas when x and y are adjacent, S + x or S + y is rainbow and x and y belong to different components of(G-xy)-S. For a connected graph G, the rainbow vertex-disconnection number of G, denoted by rvd(G), is the minimum number of colors that are needed to make G rainbow vertexdisconnected. In this paper, we characterize all graphs of order n with rainbow vertex-disconnection number k for k ∈ {1, 2, n}, and determine the rainbow vertex-disconnection numbers of some special graphs. Moreover, we study the extremal problems on the number of edges of a connected graph G with order n and rvd(G) = k for given integers k and n with 1 ≤ k ≤ n.     

Neighbor sum distinguishing total colorings of K 4-minor free graphs

A total [k]-coloring of a graph G is a mapping φ： V（G） U E（G） →{1, 2, ..., k} such that any two adjacent elements in V（G）UE（G） receive different colors. Let f（v） denote the sum of the colors of a vertex v and the colors of all incident edges of v. A total [k]-neighbor sum distinguishing-coloring of G is a total [k]-coloring of G such that for each edge uv E E（G）, f（u） ≠ f（v）. By tt [G, Xsd（ J, we denote the smallest value k in such a coloring of G. Pilniak and Woniak conjectured X＇sd（G） 〈 A（G） ＋ 3 for any simple graph with maximum degree A（G）. This conjecture has been proved for complete graphs, cycles, bipartite graphs, and subcubic graphs. In this paper, we prove that it also holds for Ka-minor free graphs. Furthermore, we show that if G is a Ka-minor flee graph with A（G） 〉 4, then ＂ Xnsd（G） 〈 A（G） ＋ 2. The bound A（G） ＋ 2 is sharp.     

A result on quasi k-connected graphs

Let G be a k-connected graph, and T be a subset of V(G)If G- T is not connected,then T is said to be a cut-set of GA k-cut-set T of G is a cut-set of G with |T | = kLet T be a k-cut-set of a k-connected graph GIf G- T can be partitioned into subgraphs G1 and G2 such that |G1| ≥ 2, |G2| ≥ 2, then we call T a nontrivial k-cut-set of GSuppose that G is a(k- 1)-connected graph without nontrivial(k- 1)-cut-setThen we call G a quasi k-connected graphIn this paper, we prove that for any integer k ≥ 5, if G is a k-connected graph without K-4, then every vertex of G is incident with an edge whose contraction yields a quasi k-connected graph, and so there are at least|V(G)|2edges of G such that the contraction of every member of them results in a quasi k-connected graph.     

Weight Choosability of Graphs with Maximum Degree 4

Let k be a positive integer. A graph G is k-weight choosable if, for any assignment L(e)of k real numbers to each e ∈ E(G), there is a mapping f : E(G) → R such that f(uv) ∈ L(uv) and ■ for each uv ∈ E(G), where ?(v) is the set of edges incident with v. As a strengthening of the famous 1-2-3-conjecture, Bartnicki, Grytczuk and Niwcyk [Weight choosability of graphs. J. Graph Theory, 60, 242–256(2009)] conjecture that every graph without isolated edge is 3-weight choosable. This conjecture is wildly open and it is even unknown whether there is a constant k such that every graph without isolated edge is k-weight choosable. In this paper, we show that every connected graph of maximum degree 4 is 4-weight choosable.     

Bifurcations of limit cycles from a quintic Hamiltonian system with a heteroclinic cycle

In this paper,we consider Li′enard systems of the form dx/dt=y,dy/dt=x+bx3-x5+ε(α+βx2+γx4)y,where b∈R,0|ε|1,(α,β,γ)∈D∈R3 and D is bounded.We prove that for |b|1(b0) the least upper bound of the number of isolated zeros of the related Abelian integrals I(h)=∮Γh(α+βx2+γx4)ydx is 2(counting the multiplicity) and this upper bound is a sharp one.     

Blow-up Sets to a Coupled Heat System

This paper deals with a heat system coupled via local and localized sources subject to null Dirichlet boundary conditions. In a previous paper of the authors, a complete result on the multiple blow-up rates was obtained. In the present paper, we continue to consider the blow-up sets to the system via a complete classification for the nonlinear parameters. That is the discussion on single point versus total blow-up of the solutions. It is mentioned that due to the influence of the localized sources, there is some substantial difficulty to be overcomed there to deal with the single point blow-up of the solutions.     

On a criterion of universal spectra

For positive integers N1 , N2 , . . . , Nn , let L = Zn + A and Λ = N1Z ×···× NnZ + Γ be two rational periodic sets, where A(1/N1 )Z ×···× (1/Nn)Z and ΓZn are finite sets with (A-A) ∩ Zn = {0} and (Γ-Γ) ∩ (N1Z ×···× NnZ) = {0}. In this note, we shall determine conditions under which the tiling set L has universal spectrum Λ. We first obtain a criterion of universal spectra. This criterion combined with the properties of compatible pair yields many necessary and sufficient conditions for Λ to be a universal spectrum for L. We then show that, under some mild additional conditions, the conjecture of Lagarias and Szabó is true. The results here extend the corresponding results of Lagarias, Szabó and Wang.     

非负对称符号模式的惯量集(英文)

For a symmetric sign pattern S1 the inertia set of S is defined to be the set of all ordered triples si（S） = {i（A） ： A = A^T ∈ Q（S）} Consider the n × n sign pattern Sn, where Sn is the pattern with zero entry （i,j） for 1 ≤ i = j ≤ n or｜i -j｜=n- 1 and positive entry otherwise. In this paper, it is proved that si（Sn） = {（n1, n2, n - n1 - n2）｜n1≥ 1 and n2 ≥ 2} for n ≥ 4.     

Certain Pair of Derivations on a Triangular Algebra

By using properties of triangular algebra, we prove that if derivations D and G on a triangular algebra T satisfy certain generalized identities, then both D and G are zero mappings. As a corollary we get that if D and G are cocentralizing on T, then both D and G are zero mappings.     

H（2）-unknotting Number of a Knot

An H（2）-move is a local move of a knot which is performed by adding a half-twisted band. It is known an H（2）-move is an unknotting operation. We define the H（2）-unknotting number of a knot K to be the minimum number of H（2）-moves needed to transform K into a trivial knot. We give several methods to estimate the H（2）-unknotting number of a knot. Then we give tables of H（2）-unknotting numbers of knots with up to 9 crossings.     

The product of the restrained domination numbers of a graph and its complement

Let G=(V,E) be a graph.A set S■V is a restrained dominating set if every vertex in V-S is adjacent to a vertex in S and to a vertex in V-S.The restrained domination number of G,denoted γr(G),is the smallest cardinality of a restrained dominating set of G.In this paper,we show that if G is a graph of order n≥4,then γr(G)γr(G)≤2n.We also characterize the graphs achieving the upper bound.     

Bifurcation in a Class of Planar Piecewise Smooth Systems with 3-parameters

This paper is concerned with the bifurcation properties on the line of discontinuity of planar piecewise smooth systems. The existence of equilibria and periodic solutions with sliding motion in a class of planar piecewise smooth systems with 3-parameters is investigated in this paper using the theory of differential inclu-sion and tools of Poincar′e maps.     

Some Results on the Polar Derivative of a Polynomial

Let P(z) be a polynomial of degree n and for any complex number α, let Dα P(z) = n P(z)+(α-z)P′(z) denote the polar derivative of P(z) with respect to α. In this paper, we obtain certain inequalities for the polar derivative of a polynomial with restricted zeros. Our results generalize and sharpen some well-known polynomial inequalities.     

Stationary Solutions for a Generalized Kadomtsev-Petviashvili Equation in Bounded Domain

In this work, we are mainly concerned with the existence of stationary solutions for a generalized Kadomtsev-Petviashvili equation in bounded domain of Rn. We utilize variational method and critical point theory to establish our main results.