The following conjecture of Alter and Wang is proven. Consider the intersection graph Gn,m,n?2m, determined by the family of all m-element subsets of an n-element set. Then any realization of Gn,m as an intersection graph by a family of sets satisfies |∪iAi|?n; and if |∪iAi|=n, then F must be the family of all m-element subsets of ∪iAi. 相似文献
For graphs G and F we write F → (G)1r if every r-coloring of the vertices of F results in a monochromatic copy of G. The global density m(F) of F is the maximum ratio of the number of edges to the number of vertices taken over all subgraphs of F. Let We show that The lower bound is achieved by complete graphs, whereas, for all r ≥ 2 and ? > 0, mcr(Sk, r) > r - ? for sufficiently large k, where Sk is the star with k arms. In particular, we prove that 相似文献
For a graph G, a subset of vertices D is a dominating set if for each vertex X not in D, X is adjacent to at least one vertex of D. The domination number, γ(G), is the order of the smallest such set. An outstanding conjecture in the theory of domination is for any two graph G and H, One result presented in this paper settles this question in the case when at least one of G or H is a tree. We show that for all graphs G and any tree T. Furthermore, we supply a partial characterization for which pairs of trees, T1 and T2, strict inequality occurs. We show for almost all pairs of trees. 相似文献
For given graphs G and H, the Ramsey number R(G,H) is the smallest natural number n such that for every graph F of order n: either F contains G or the complement of F contains H. In this paper we investigate the Ramsey number of a disjoint union of graphs . For any natural integer k, we contain a general upper bound, R(kG,H)?R(G,H)+(k-1)|V(G)|. We also show that if m=2n-4, 2n-8 or 2n-6, then R(kSn,Wm)=R(Sn,Wm)+(k-1)n. Furthermore, if |Gi|>(|Gi|-|Gi+1|)(χ(H)-1) and R(Gi,H)=(χ(H)-1)(|Gi|-1)+1, for each i, then . 相似文献
We show that the vertex set of any graph G with p?2 vertices can be partitioned into non-empty sets V1, V2, such that the maximum degree of the induced subgraph 〈Vi〉 does not exceed where pi = |Vi|, for i=1, 2. Furthermore, the structure of the induced subgraphs is investigated in the extreme case. 相似文献
Let k be an arbitrary field, X1,….,Xn indeterminates over k and F1…, F3 ε ∈ k[X1…,Xn] polynomials of maximal degree $ d: = \mathop {\max }\limits_{1 \le i \le a} \deg $ (Fi). We give an elementary proof of the following effective Nullstellensatz: Assume that F1,…,F have no common zero in the algebraic closure of k. Then there exist polynomials P1…, P3 ε ∈ k[X1…,Xn] such that $ 1: = \mathop \Sigma \limits_{1 \le i \le a} $ PiFi and This result has many applications in Computer Algebra. To exemplify this, we give an effective quantitative and algorithmic version of the Quillen-Suslin Theorem baaed on our effective Nullstellensatz. 相似文献
When can a k-edge-coloring of a subgraph K of a graph G be extended to a k-edge-coloring of G? One necessary condition is that for all X ? E(G) - E(K), where μi(X) is the maximum cardinality of a subset of X whose union with the set of edges of K colored i is a matching. This condition is not sufficient in general, but is sufficient for graphs of very simple structure. We try to locate the border where sufficiency ends. 相似文献
Some laws in physics describe the change of a flux and are represented by parabolic equations of the form (*) \documentclass{article}\pagestyle{empty}\begin{document}$$\frac{{\partial u}}{{\partial t}}=\frac{\partial}{{\partial x_j }}(\eta \frac{{\partial u}}{{ax_j}}-vju),$$\end{document} j≤m, where η and vj are functions of both space and time. We show under quite general assumptions that the solutions of equation (*) with homogeneous Dirichlet boundary conditions and initial condition u(x, 0) = uo(x) satisfy The decay rate d > 0 only depends on bounds for η, v and G § Rm the spatial domain, while the constant c depends additionally on which norm is considered. For the solutions of equation (*) with homogeneous Neumann boundary conditions and initial condition u0(x) ≥ 0 we derive bounds d1u1 ≤ u(x, t) ≤ d2u2, Where di, i = 1, 2, depend on bounds for η, v and G, and the ui are bounds on the initial condition u0. 相似文献