Showing papers in "Combinatorica in 1983"
TL;DR: Several theorems involving configurations of points and lines in the Euclidean plane are established, including one that shows that there is an absolute constantc3 so that whenevern points are placed in the plane not all on the same line, then there is one point on more thanc3n of the lines determined by then points.
Abstract: In this paper, we establish several theorems involving configurations of points and lines in the Euclidean plane. Our results answer questions and settle conjectures of P. Erdos, G. Purdy, and G. Dirac. The principal result is that there exists an absolute constantc
1 so that when
$$\sqrt n \leqq t \leqq \left( {_2^n } \right)$$
, the number of incidences betweenn points andt lines is less thanc
1
n
2/3
t
2/3. Using this result, it follows immediately that there exists an absolute constantc
2 so that ifk≦√n, then the number of lines containing at leastk points is less thanc
2
n
2/k
3. We then prove that there exists an absolute constantc
3 so that whenevern points are placed in the plane not all on the same line, then there is one point on more thanc
3
n of the lines determined by then points. Finally, we show that there is an absolute constantc
4 so that there are less than exp (c
4 √n) sequences 2≦y
1≦y
2≦...≦y
r for which there is a set ofn points and a setl
1,l
2, ...,l
t
oft lines so thatl
j
containsy
j
points.
644 citations
TL;DR: A sorting network withcn logn comparisons where in thei-th step of the algorithm the contents of registersRj, andRk, wherej, k are absolute constants then change their contents or not according to the result of the comparison.
Abstract: We give a sorting network withcn logn comparisons. The algorithm can be performed inc logn parallel steps as well, where in a parallel step we comparen/2 disjoint pairs. In thei-th step of the algorithm we compare the contents of registersR
j(i)
, andR
k(i)
, wherej(i), k(i) are absolute constants then change their contents or not according to the result of the comparison.
497 citations
TL;DR: The basic question investigated here is: At least how many copies ofL ε ℒ must occur in a graphGn onn vertices with ex (n,ℒ)+k edges (hyperedges)?
Abstract: We shall consider graphs (hypergraphs) without loops and multiple edges. Let ℒ be a family of so called prohibited graphs and ex (n, ℒ) denote the maximum number of edges (hyperedges) a graph (hypergraph) onn vertices can have without containing subgraphs from ℒ. A graph (hyper-graph) will be called supersaturated if it has more edges than ex (n, ℒ). IfG hasn vertices and ex (n, ℒ)+k edges (hyperedges), then it always contains prohibited subgraphs. The basic question investigated here is: At least how many copies ofL e ℒ must occur in a graphG
n
onn vertices with ex (n, ℒ)+k edges (hyperedges)?
287 citations
TL;DR: The Erdős-Ko-Rado theorem implies that f(n, k, s) is the maximum cardinality of a family of k-subsets of ann-set as mentioned in this paper.
Abstract: Let ℱ be a family ofk-subsets of ann-set. Lets be a fixed integer satisfyingk≦s≦3k. Suppose that forF
1,F
2,F
3 ∈ ℱ |F
1 ∪F
2 ∪F
3|≦s impliesF
1 ∩F
2 ∩F
3 ≠ 0. Katona asked what is the maximum cardinality,f(n, k, s) of such a system. The Erdős-Ko-Rado theorem impliesf(n, k, s)=
$$\left( {_{k - 1}^{n - 1} } \right)$$
fors=3k andn≧2k. In this paper we show thatf(n, k, s)=
$$\left( {_{k - 1}^{n - 1} } \right)$$
holds forn>n
0(k) if and only ifs≧2k. Equality holds only if every member of ℱ contains a fixed element of the underlying set. Further we solve the problem fork=3,s=5,n≧3000. This result sharpens a theorem of Bollobas.
162 citations
TL;DR: It will be shown here that for some point of S the number of connecting lines through it exceedsc · n, and the following conjecture of Erdős is proved: If any straight line contains at mostn−x points of S, then the numberof connecting lines determined byS is greater thanc · x · n.
Abstract: LetS be a set ofn non-collinear points in the Euclidean plane. It will be shown here that for some point ofS the number ofconnecting lines through it exceedsc · n. This gives a partial solution to an old problem of Dirac and Motzkin. We also prove the following conjecture of Erdős: If any straight line contains at mostn−x points ofS, then the number of connecting lines determined byS is greater thanc · x · n.
128 citations
TL;DR: It is proved that the length of the period of a simple model of society with a symmetric functionw(u, v) measuring the influence of the opinion of memberv on that of memberu is either one or two.
Abstract: We propose a simple model of society with a symmetric functionw(u, v) measuring the influence of the opinion of memberv on that of memberu. The opinions are chosen from a finite set. At each step everyone accepts the majority opinion (with respect tow) of the other members. The behaviour of such a society is clearly periodic after some initial time. We prove that the length of the period is either one or two.
109 citations
TL;DR: This paper generalizes and investigates the case when C2k, a cycle of 2k vertices, is forbidden and the maximum number of edges a graphGn of ordern can have without containing subgraphs from ℒ.
Abstract: Given a class ℒ of (so called “forbidden”) graphs, ex (n, ℒ) denotes the maximum number of edges a graphG n of ordern can have without containing subgraphs from ℒ. If ℒ contains bipartite graphs, then ex (n, ℒ)=O(n 2−c ) for somec>0, and the above problem is calleddegenerate. One important degenerate extremal problem is the case whenC 2k , a cycle of 2k vertices, is forbidden. According to a theorem of P. Erdős, generalized by A. J. Bondy and M. Simonovits [32, ex (n, {C 2k })=O(n 1+1/k ). In this paper we shall generalize this result and investigate some related questions.
98 citations
TL;DR: It is proved that a non-bipartite matching-covered graph contains K4 or K2⊕K3 (the triangular prism).
Abstract: We call a graphmatching-covered if every line belongs to a perfect matching. We study the technique of “ear-decompositions” of such graphs. We prove that a non-bipartite matching-covered graph containsK
4 orK
2⊕K
3 (the triangular prism). Using this result, we give new characterizations of those graphs whose matching and covering numbers are equal. We apply these results to the theory of τ-critical graphs.
96 citations
TL;DR: The main object of this paper is to describe the behavior of the functionf(s,t;k), usually thinking ofs andt fixed, and lettingk become large.
Abstract: Letf(s, t; k) be the largest value ofm such that it is possible tok-color the edges of the complete graphK
m
so that everyK
s
⊆K
m
has exactlyt colors occuring on its edges. The main object of this paper is to describe the behavior of the functionf(s,t;k), usually thinking ofs andt fixed, and lettingk become large.
92 citations
TL;DR: It is shown that, for each natural numberk, these exists a (smallest) natural numberf(k) such that any digraph of minimum outdegree at leastf( k) containsk disjoint cycles.
Abstract: We show that, for each natural numberk, these exists a (smallest) natural numberf(k) such that any digraph of minimum outdegree at leastf(k) containsk disjoint cycles. We conjecture thatf(k)=2k−1 and verify this fork=2 and we show that, for eachk≧3, the determination off(k) is a finite problem.
76 citations
TL;DR: Some techniques for nonlinear systematic error-correcting codes over the two-element alphabet are introduced and new perfect 1-error-Correcting (15,11)-codes are constructed.
Abstract: We introduce some techniques for nonlinear systematic error-correcting codes over the two-element alphabet. As an application, we construct new perfect 1-error-correcting (15,11)-codes.
TL;DR: The paper deals with common generalizations of classical results of Ramsey and Turán with one of the main results Assumek≧2, ε>0,Gn is a sequence of graphs ofn-vertices and at least 1/2((3k−5) / (3k+2)+ε)n2 edges, and the size of the largest independent set inGn iso(n).
Abstract: The paper deals with common generalizations of classical results of Ramsey and Turan. The following is one of the main results. Assumek≧2, e>0,G
n
is a sequence of graphs ofn-vertices and at least 1/2((3k−5) / (3k−2)+e)n
2 edges, and the size of the largest independent set inG
n
iso(n). LetH be any graph of arboricity at mostk. Then there exists ann
0 such that allG
n
withn>n
0 contain a copy ofH. This result is best possible in caseH=K
2k
.
TL;DR: This paper proves that to each vertex of the graph a string ofn−1 symbols from the alphabet {0, 1, *} can be assigned in such a way that the length of the shortest path between two vertices is equal to the Hamming distance between the corresponding strings.
Abstract: We prove a conjecture of R. L. Graham and H. O. Pollak to the effect that every connected graph onn vertices has a distance-preserving embedding in “squashed cube” of dimensionn−1. This means that to each vertex of the graph a string ofn−1 symbols from the alphabet {0, 1, *} can be assigned in such a way that the length of the shortest path between two vertices is equal to the Hamming distance between the corresponding strings, with * being regarded as at distance zero from either 1 or 0. Our labelling thus provides an efficient addressing scheme for the loop-switching communications system proposed by J. R. Pierce.
TL;DR: It is shown how to construct cubic graphs which have automorphism groups acting regularly on thes-arcs,s=4 or 5.
Abstract: We show how to construct cubic graphs which have automorphism groups acting regularly on thes-arcs,s=4 or 5.
TL;DR: A decomposition theory for submodular functions is described, and any such function is shown to have a unique decomposition consisting of indecomposable functions and certain highly decomposability functions, and the latter are completely characterized.
Abstract: A decomposition theory for submodular functions is described. Any such function is shown to have a unique decomposition consisting of indecomposable functions and certain highly decomposable functions, and the latter are completely characterized. Applications include decompositions of hypergraphs based on edge and vertex connectivity, the decomposition of matroids based on three-connectivity, the Gomory—Hu decomposition of flow networks, and Fujishige’s decomposition of symmetric submodular functions. Efficient decomposition algorithms are also discussed.
TL;DR: Greedoids were introduced by the authors as generalizations of matroids providing a framework for the greedy algorithm from a structural aspect and it is shown that Rado’s theorem on independent transversals of subsets ofMatroids remains valid for feasible transversal of certain sets of greedoids.
Abstract: Greedoids were introduced by the authors as generalizations of matroids providing a framework for the greedy algorithm. In this paper they are studied from a structural aspect. Definitions of basic matroid-theoretical concepts such as rank and closure can be generalized to greedoids, even though they loose some of their fundamental properties. The rank function of a greedoid is only “locally” submodular. The closure operator is not monotone but possesses a (relaxed) Steinitz—McLane exchange property. We define two classes of subsets, called rank-feasible and closure-feasible, so that the rank and closure behave nicely for them. In particular, restricted to rank-feasible sets the rank function is submodular. Finally we show that Rado’s theorem on independent transversals of subsets of matroids remains valid for feasible transversals of certain sets of greedoids.
TL;DR: It is proved, that f(s, t) ≦ 4s+4t − 13 and a similar problem for the minimum degree is examined.
Abstract: C. Thomassen and M. Szegedy proved the existence of a functionf(s, t) such that the points of anyf(s, t)-connected graph have a decomposition into two non-empty sets such that the subgraphs induced by them ares-connected andt-connected, respectively. We prove, thatf(s, t) ≦ 4s+4t − 13 and examine a similar problem for the minimum degree.
TL;DR: The problem of finding a perfect 2-matching of a graph such that any cycle in the solution contains more thank edges was studied in this article, where it was shown that the problem can be solved in polynomial time.
Abstract: A(perfect) 2-matching in a graphG=(V, E) is an assignment of an integer 0, 1 or 2 to each edge of the graph in such a way that the sum over the edges incident with each node is at most (exactly) two The incidence vector of a Hamiltonian cycle, if one exists inG, is an example of a perfect 2-matching Fork satisfying 1≦k≦|V|, we letPk denote the problem of finding a perfect 2-matching ofG such that any cycle in the solution contains more thank edges We call such a matching aperfect Pk-matching Then fork
TL;DR: In this paper, the authors make geometers and combinatorialists familiar with old and new connections between the geometry of Lorentz space and the combinatorics of graph theory.
Abstract: The aim of the paper is to make geometers and combinatorialists familiar with old and new connections between the geometry of Lorentz space and combinatorics. Among the topics treated are equiangular lines and their relations to spherical 2-distance sets; spherical and hyperbolic root systems and their relation to graphs whose second largest eigenvalue does not exceed one or two, respectively; and work of Niemeier, Vinberg, Conway and Sloane on Euclidean and Lorentzian unimodular lattices.
TL;DR: In this article, the authors locate a sharp threshold for the property of having a hamiltonian path in the scale of random graph spaces, where the ap-band across which the asymptotic probability of Q jumps from 0 to 1 is defined.
Abstract: A threshold for a graph propertyQ in the scale of random graph spacesG
n,p
is ap-band across which the asymptotic probability ofQ jumps from 0 to 1. We locate a sharp threshold for the property of having a hamiltonian path.
TL;DR: In this article, the authors studied the matroid of functions of strength on a finite set X, where a function f:P(X) → Z is said to have strength if and only if:
Abstract: For a finite setX, a functionf:P(X) →Z is said to have strengtht if
$$\sum\limits_{A\underline{\underline \subset } B} {f(B) = 0} $$
for allA ∈P (X), |A|≦t. Supports of functions of strengtht define a matroid onP(X). We study the circuits in this matroid. Some other related problems are also discussed.
TL;DR: Lower and upper estimates on maxm(F) are given where max runs over the set of all closure operations onn elements.
Abstract: A matrixM withn columns represents a closure operationF(A), (A⊂X, |X|=n) if for anyA, any two rows equal in the columns corresponding toA are also equal inF(A). Letm(F) be the minimum number of rows of the matrices representingF. Lower and upper estimates on maxm(F) are given where max runs over the set of all closure operations onn elements.
TL;DR: The main result of this paper is an algorithm that shows that a simple hypergraph on at mostm vertices with no special cycles can be completed (by adding edges) to a solution.
Abstract: A special cycle in a hypergraph is a cyclex1E1x2E2x3 ...xnEnx1 ofn distinct verticesxi andn distinct edgesEj (n≧3) whereEi∩{x1,x2, ...,xn}={xi,xi+1} (xn+1=x1). In the equivalent (0, 1)-matrix formulation, a special cycle corresponds to a square submatrix which is the incidence matrix of a cycle of size at least 3. Hypergraphs with no special cycles have been called totally balanced by Lovasz. Simple hypergraphs with no special cycles onm vertices can be shown to have at most (2m)+m+1 edges where the empty edge is allowed. Such hypergraphs with the maximum number of edges have a fascinating structure and are called solutions. The main result of this paper is an algorithm that shows that a simple hypergraph on at mostm vertices with no special cycles can be completed (by adding edges) to a solution.
TL;DR: It is proved that the largestm such that there is a digraph onn-vertices andm edges which is a subgraph of every tournament on n- vertices is a spanning weakly connected digraph.
Abstract: Letf(n) (resp.g(n)) be the largestm such that there is a digraph (resp. a spanning weakly connected digraph) onn-vertices andm edges which is a subgraph of every tournament onn-vertices. We prove that
$$n\log _2 n - c_1 n \geqq f(n) \geqq g(n) \geqq n\log _2 n - c_2 n\log \log n.$$
TL;DR: It is shown that each loopless 2k-regular undirected graph onn vertices has at least two eulerian orientations and at most three, and that, for each fixedk, these ground numbers are best possible.
Abstract: We show that each loopless 2k-regular undirected graph onn vertices has at least\(\left( {2^{ - k} \left( {_k^{2k} } \right)} \right)^n \) and at most\(\sqrt {\left( {_k^{2k} } \right)^n } \) eulerian orientations, and that, for each fixedk, these ground numbers are best possible.
TL;DR: Methods are presented to show that various hypergraph families are τ-bound and/or ϱ-bound, and to families defined by hypergraph theoretic terms like the family where every subhypergraph has the Helly-property.
Abstract: The transversal number, packing number, covering number and strong stability number of hypergraphs are denoted by τ, ν, ϱ and α, respectively. A hypergraph family þ is called τ-bound (ϱ-bound) if there exists a “binding function”f(x) such that τ(H)≦f(v(H)) (ϱ(H)≦f(α(H))) for allH ∈ þ. Methods are presented to show that various hypergraph families are τ-bound and/or ϱ-bound. The results can be applied to families of geometrical nature like subforests of trees, boxes, boxes of polyominoes or to families defined by hypergraph theoretic terms like the family where every subhypergraph has the Helly-property.
TL;DR: In this article, the best possible bounds for the number of edges in a graph which is forced to be contained in every graph onn vertices ande edges were obtained. But these bounds are asymptotically best possible.
Abstract: How many edges can be in a graph which is forced to be contained in every graph onn vertices ande edges? In this paper we obtain bounds which are in many cases asymptotically best possible.
TL;DR: A purely deterministic proof of the following theorem of J. Komlós and M. Sulyok is given: one can multiply some rows and columns by −1 such that the absolute value of the sum of the elements of the matrix is ≦2 ifn is even and 1 if n is odd.
Abstract: Let an arbitrary matrix A = (a ij), 1 ≤ i ≤ K, 1 ≤ j ≤ L be given with all |a ij| ≤ 1. By a row shift we mean the act of replacing, for a particular i, all coefficients a ij in the i-th row by their negatives (-a ij). A column shift is defined similarly. A line shift denotes either a row or a column shift. Consider the following solitaire game. The player applies a succession of line shifts to A. His object is to make the absolute value of the sum of all the coefficients of A (which we shall denote by |A|) as small as possible. We shall show (answering a question of J. Komlos) that the player can always make |A| ≤ c 0 where c 0 is an absolute constant — i.e., independent of K, L, and the initial matrix. We make no attempt to find the minimal possible c 0.
TL;DR: It is shown that any graph of uncountable chromatic number contains an edge through which there are cycles of all (but finitely many) lengths.
Abstract: We answer a question of Erdős [1], [2] by showing that any graph of uncountable chromatic number contains an edge through which there are cycles of all (but finitely many) lengths.
TL;DR: This paper investigates to what extent, the Milliken partition theorem for finitistic trees is a density result.
Abstract: This paper investigates to what extent, the Milliken partition theorem for finitistic trees is a density result.