共查询到20条相似文献,搜索用时 15 毫秒
1.
Sukhamay Kundu 《Journal of Combinatorial Theory, Series B》1974,17(2):199-203
It is shown that an n-edge connected graph has at least ?? pairwise edge-disjoint spanning trees. This bound is best possible in general. A maximal planar graph with four or more vertices contains two edge-disjoint spanning trees. For a maximal toroidal graph, this number is three. 相似文献
2.
Ronald L. Rivest 《Discrete Mathematics》1977,17(2):181-186
We consider the minimax number of questions required to determine which leaf in a finite binary tree T your opponent has chosen, where each question may ask if the leaf is in a specified subtree of T. The requisite number of questions is shown to be approximately the logarithm (base &0slash;) of the number of leaves in T as T becomes large, where Ø = 1.61803… is the “golden ratio”. Specifically, q questions are sufficient to reduce the number of possibilities by a factor of (where F, is the ith Fibonacci number), and this is the best possible. 相似文献
3.
R Kemp 《Journal of Combinatorial Theory, Series B》1983,34(2):191-208
The number of planted plane trees with n nodes, m leaves, and height h is computed. Assuming that all n-node planted plane trees with m leaves are equally likely, it is shown that the average height is asymptotically given for all ε > 0 and fixed , 0 < ? < 1, by 相似文献
4.
Peter Frankl 《Journal of Combinatorial Theory, Series A》1983,34(1):41-45
For a family of subsets of an n-set X we define the trace of it on a subset Y of X by T(Y) = {F∩Y:F?}. We say that (m,n) → (r,s) if for every with | we can find a Y?X|Y| = s such that |T(Y)| ? r. We give a unified proof for results of Bollobàs, Bondy, and Sauer concerning this arrow function, and we prove a conjecture of Bondy and Lovász saying , which generalizes Turán's theorem on the maximum number of edges in a graph not containing a triangle. 相似文献
5.
Shawpawn Kumar Das 《Journal of Combinatorial Theory, Series B》1973,15(2):184-199
A connected topology is said to be maximal connected if strictly finer than implies that is disconnected. In this paper, it is shown that the number of homeomorphism classes of maximal connected topologies defined on a set with n points is equal to twice the number of n point trees minus the number of n point trees possessing a symmetry line. An enumeration of a class called critical connected topologies, which includes the maximal connected spaces is then carried out with the help of Pólya's theorem. Another result is that a chain of connected n point T0 topologies, linearly ordered by strict fineness, can contain a maximum of topologies, and, moreover, this number is the best possible upper bound for the length of such a chain. 相似文献
6.
Denote by λ2(T) the second largest eigenvalue of a tree T. An easy algorithm is given to decide whether λ2(T)?λ for a given number λ, and a structure theorem for trees withλ2(T)?λ is proved. Also, it is shown that a tree T with n vertices has ; this bound is best possible for odd n. 相似文献
7.
For a permutation σ of the integers from 1 to n, let ?(σ) be the smallest number of prefix reversals that will transform σ to the identity permutation, and let ?(n) be the largest such ?(σ) for all σ in (the symmetric group) Sn. We show that , and that for n a multiple of 16. If, furthermore, each integer is required to participate in an even number of reversed prefixes, the corresponding function g(n) is shown to obey . 相似文献
8.
Bondy conjectured [1] that: if G is a k-connected graph, where k ≥ 2, such that the degree-sum of any k + 1 independent vertices is at least m, then G contains a cycle of length at least: (n denotes the order of G). We prove here that this result is true. 相似文献
9.
H.S Witsenhausen 《Journal of Combinatorial Theory, Series A》1974,17(3):387-390
It is established that the maximum number of points required to puncture 3n sets of probability is 2n, as had been conjectured. In fact, for 1 ≤ m ≤ n, a family of m sets of probability can be punctured using no more than points. The more general statement that (k + 1)n sets of probability require at most 2n points for puncturing is shown to be false already for k = 3. 相似文献
10.
Peter Kirschenhofer 《Discrete Applied Mathematics》1984,7(2):161-181
Let k denote one of the families of binary trees, t-aray trees (t> 2) or ordered trees with nodes labelled monotonically by elements of {1 < 2 < ? < k}. The average height of the j-th leaf of the trees of k with exactly n nodes is shown to converge to a finite limit (depending on k and j) for n → ∞. The limit is determined explicitly for small values of k and its asymptotic behaviour in j and k is investigated. Some recent results on the average shape of rooted tree structures appear as special cases. 相似文献
11.
For finite graphs F and G, let NF(G) denote the number of occurrences of F in G, i.e., the number of subgraphs of G which are isomorphic to F. If and are families of graphs, it is natural to ask then whether or not the quantities NF(G), F∈, are linearly independent when G is restricted to . For example, if = {K1, K2} (where Kn denotes the complete graph on n vertices) and is the family of all (finite) trees, then of course NK1(T) ? NK2(T) = 1 for all T∈. Slightly less trivially, if = {Sn: n = 1, 2, 3,…} (where Sn denotes the star on n edges) and again is the family of all trees, then Σn=1∞(?1)n+1NSn(T)=1 for all T∈. It is proved that such a linear dependence can never occur if is finite, no F∈ has an isolated point, and contains all trees. This result has important applications in recent work of L. Lovász and one of the authors (Graham and Lovász, to appear). 相似文献
12.
A.H Zemanian 《Journal of Mathematical Analysis and Applications》1976,55(2):394-406
The subject of this work is the voltage and current transfer ratios of three-terminal networks having no mutual coupling and whose impedances are analytic functions taking their values in an abelian self-adjoint algebra of bounded linear operators on a Hilbert space. Each such value is also assumed to be invertible. It is shown that these ratios have the form [I + A(ζ)]?1, where, for each ζ in a sufficiently small open cone in the right-half complex plane with apex at the origin and the real axis as its bisector, the numerical range of A(ζ) is contained in a compact subset of the open right-half plane. This implies that the ratios are strictly contractive for each ζ in the cone. The angle of the cone is , where k is the number of internal nodes of a certain “surrogate” network; this result is best possible. For two-terminal-pair networks the ratios are shown to be strictly contractive for each ζ in a similar cone with angle . 相似文献
13.
Jerrold R. Griggs 《Discrete Mathematics》1979,28(1):37-47
It is shown that the interval number of a graph on n vertices is at most , and this bound is best possible. This means that we can represent any graph on n vertices as an intersection graph in which the sets assigned to the vertices each consist of the union of at most finite closed intervals on the real line. This upper bound is attained by the complete bipartite graph K[n/2],[n/2]. 相似文献
14.
Shawpawn Kumar Das 《Journal of Combinatorial Theory, Series B》1979,26(3):295-299
Let be a finite topology. If P and Q are open sets of (Q may be the null set) then P is a minimal cover of Q provided Q ? P and there does not exist any open set R of such that Q ? R ? P. A subcollection of the open sets of is termed an i-discrete collection of provided contains every open O ∈ with the property that ? ? O ? ? , contains exactly i minimal covers of ? , and provided ? = ?{O | O ∈ and O is a minimal cover of ? }. A single open set is a O-discrete collection. The number of distinct i-discrete collections of is denoted by p(, i). If there does not exist any i-discrete collection then p(,i) = 0, and this happens trivially for the case when i is greater than the number of points on which is defined. The object of this article is to establish the theorem: For any finite topology , the quantity E() = Σi = 0∞ (?1)ip(, i) = 1. 相似文献
15.
W Fernandez de la Vega 《Journal of Combinatorial Theory, Series B》1983,35(3):328-332
For any tournament T on n vertices, let h(T) denote the maximum number of edges in the intersection of T with a transitive tournament on the same vertex set. Sharpening a previous result of Spencer, it is proved that, if Tn denotes the random tournament on n vertices, then, as n → ∞. 相似文献
16.
Let be the number of conjugacy classes of elements in SL(2, ) with trace t, and let be the number of equivalence classes of binary quadratic forms with discriminant n. Then for t≠±2, . For all real θ > 0 there is a T(θ) such that whenever |t|>T(θ), . There is a c>0 such that for those t such that t2?4 is squarefree, . 相似文献
17.
M.-C. Heydemann 《Discrete Mathematics》1982,41(3):241-251
In this article, we give conditions on the total degrees of the vertices in a strong digraph implying the existence of a cycle of length at least , where n is the number of vertices of the graph and h an integer, 1?h?n?1. The same conditions imply the existence of a path of length . In the case of strong oriented graphs (antisymmetric digraphs) we improve these conditions. In both cases, we show that the given conditions are the best possible. 相似文献
18.
We define two two-variable polynomials for rooted trees and one two-variable polynomial for unrooted trees, all of which are based on the coranknullity formulation of the Tutte polynomial of a graph or matroid. For the rooted polynomials, we show that the polynomial completely determines the rooted tree, i.e., rooted trees T1 and T2 are isomorphic if and only if f(T1) = f(T2). The corresponding question is open in the unrooted case, although we can reconstruct the degree sequence, number of subtrees of size k for all k, and the number of paths of length k for all k from the (unrooted) polynomial. The key difference between these three polynomials and the standard Tutte polynomial is the rank function used; we use pruning and branching ranks to define the polynomials. We also give a subtree expansion of the polynomials and a deletion-contraction recursion they satisfy. 相似文献
19.
Let T be a rooted tree structure with n nodes a1,…,an. A function f: {a1,…,an} into {1 < ? < k} is called monotone if whenever ai is a son of aj, then f(ai) ≥ f(aj). The average number of monotone bijections is determined for several classes of tree structures. If k is fixed, for the average number of monotone functions asymptotic equivalents of the form c · ??nn? (n → ∞) are obtained for several classes of tree structures. 相似文献
20.
George B Purdy 《Journal of Combinatorial Theory, Series A》1974,17(1):131-133
There are at most 2n spheres tangent to all n + 1 faces of an n-simplex. It has been shown that the minimum number of such spheres is if n is odd and if n is even. The object of this note is to show that this result is a consequence of a theorem in graph theory. 相似文献