Showing papers in "Combinatorica in 1988"

The chromatic number of random graphs

Abstract: For a fixed probabilityp, 0

The gap between monotone and non-monotone circuit complexity is exponential

Abstract: A. A. Razborov has shown that there exists a polynomial time computable monotone Boolean function whose monotone circuit complexity is at leastnc losn. We observe that this lower bound can be improved to exp(cn1/6−o(1)). The proof is immediate by combining the Alon—Boppana version of another argument of Razborov with results of Grotschel—Lovasz—Schrijver on the Lovasz — capacity, ϑ of a graph.

Rubber bands, convex embeddings and graph connectivity"

Abstract: We give various characterizations ofk-vertex connected graphs by geometric, algebraic, and “physical” properties. As an example, a graphG isk-connected if and only if, specifying anyk vertices ofG, the vertices ofG can be represented by points of ℝk−1 so that nok are on a hyper-plane and each vertex is in the convex hull of its neighbors, except for thek specified vertices. The proof of this theorem appeals to physics. The embedding is found by letting the edges of the graph behave like ideal springs and letting its vertices settle in equilibrium. As an algorithmic application of our results we give probabilistic (Monte-Carlo and Las Vegas) algorithms for computing the connectivity of a graph. Our algorithms are faster than the best known (deterministic) connectivity algorithms for allk≧√n, and for very dense graphs the Monte Carlo algorithm is faster by a linear factor.

A lower bound for finding predecessors in Yao's cell probe model

Abstract: LetL be the set consisting of the firstq positive integers. We prove in this paper that there does not exist a data structure for representing an arbitrary subsetA ofL which uses poly (¦A¦) cells of memory (where each cell holdsc logq bits of information) and which the predecessor inA of an arbitraryx≦q can be determined by probing only a constant (independent ofq) number of cells. Actually our proof gives more: the theorem remains valid if this number is less thane log logq, that is D. E. Willard's algorithm [2] for finding the predecessor inO(log logq) time is optimal up to a constant factor.

On multiplicative graphs and the product conjecture

Abstract: We study the following problem: which graphsG have the property that the class of all graphs not admitting a homomorphism intoG is closed under taking the product (conjunction)? Whether all undirected complete graphs have the property is a longstanding open problem due to S. Hedetniemi. We prove that all odd undirected cycles and all prime-power directed cycles have the property. The former result provides the first non-trivial infinite family of undirected graphs known to have the property, and the latter result verifies a conjecture of Nesetřil and Pultr These results allow us (in conjunction with earlier results of Nesetřil and Pultr [17], cf also [7]) to completely characterize all (finite and infinite, directed and undirected) paths and cycles having the property. We also derive the property for a wide class of 3-chromatic graphs studied by Gerards, [5].

Random interval graphs

Abstract: In this paper we introduce a notion ofrandom interval graphs: the intersection graphs of real, compact intervals whose end points are chosen at random. We establish results about the number of edges, degrees, Hamiltonicity, chromatic number and independence number of almost all interval graphs.

Bounding the diameter of distance-regular graphs

Abstract: LetG be a connected distance-regular graph with valencyk>2 and diameterd, but not a complete multipartite graph. Suppose thatθ is an eigenvalue ofG with multiplicitym and thatθ≠±k. We prove that bothd andk are bounded by functions ofm. This implies that, ifm>1 is given, there are only finitely many connected, co-connected distance-regular graphs with an eigenvalue of multiplicitym.

Hereditary modular graphs

Abstract: In a hereditary modular graphG, for any three verticesu, v, w of an isometric subgraph ofG, there exists a vertex of this subgraph that is simultaneously on some shortestu, v-path,u, w-path andv, w-path. It is shown that the hereditary modular graphs are precisely those bipartite graphs which do not contain any isometric cycle of length greater than four. There is a polynomial-time algorithm available which decides whether a given (bipartite) graph is hereditary modular or not. Finally, the chordal bipartite graphs are characterized by forbidden isometric subgraphs.

Small topological complete subgraphs of dense graphs

Abstract: A graph ofn vertices and\(4^{t^2 } n^{1 + \varepsilon } \) edges contains aTKt on at most 7t2 logt/e vertices. This answers a question of P. Erdős.

On sums of subsets of a set of integers

Abstract: Forr≧2 letp(n, r) denote the maximum cardinality of a subsetA ofN={1, 2,...,n} such that there are noB⊂A and an integery with\(\mathop \sum \limits_{b \in B} b = y^r \)b=yr. It is shown that for anye>0 andn>n(e), (1+o(1))21/(r+1)n(r−1)/(r+1)≦p(n, r)≦nɛ+2/3 for allr≦5, and that for every fixedr≧6,p(n, r)=(1+o(1))·21/(r+1)n(r−1)/(r+1) asn→∞. Letf(n, m) denote the maximum cardinality of a subsetA ofN such that there is noB⊂A the sum of whose elements ism. It is proved that for 3n6/3+ɛ≦m≦n2/20 log2n andn>n(e), f(n, m)=[n/s]+s−2, wheres is the smallest integer that does not dividem. A special case of this result establishes a conjecture of Erdős and Graham.

Improved lower bounds on the length of Davenport-Schinzel sequences

Abstract: We derive lower bounds on the maximal lengthλs(n) of (n, s) Davenport Schinzel sequences. These bounds have the form λ2s=1(n)=Ω(nαs(n)), whereα(n) is the extremely slowly growing functional inverse of the Ackermann function. These bounds extend the nonlinear lower boundλ3(n)=Ω(nα(n)) due to Hart and Sharir [5], and are obtained by an inductive construction based upon the construction given in [5].

A random 1-011-011-01algorithm for depth first search

Abstract: In this paper we present a fast parallel algorithm for constructing a depth first search tree for an undirected graph. The algorithm is anRNC algorithm, meaning that it is a probabilistic algorithm that runs in polylog time using a polynomial number of processors on aP-RAM. The run time of the algorithm isO(T MM(n) log3 n), and the number of processors used isP MM (n) whereT MM(n) andP MM(n) are the time and number of processors needed to find a minimum weight perfect matching on ann vertex graph with maximum edge weightn.

Improved processor bounds for combinatorial problems in RNC

Abstract: Our main result improves the known processor bound by a factor ofn4 (maintaining the expected parallel running time,O(log3n)) for the following important problem:find a perfect matching in a general or in a bipartite graph with n vertices. A solution to that problem is used in parallel algorithms for several combinatorial problems, in particular for the problems of finding i) a (perfect) matching of maximum weight, ii) a maximum cardinality matching, iii) a matching of maximum vertex weight, iv) a maximums-t flow in a digraph with unit edge capacities. Consequently the known algorithms for those problems are substantially improved.

Optima of dual integer linear programs

Abstract: We consider dual pairs of packing and covering integer linear programs. Best possible bounds are found between their optimal values. Tight inequalities are obtained relating the integral optima and the optimal rational solutions.

Some Ramsey-Turán type results for hypergraphs

Abstract: To everyk-graphG letπ(G) be the minimal real numberπ such that for everye>0 andn>n0(e,G) everyk-graphH withn vertices and more than (π+e) (\(\left( {\pi + \varepsilon } \right)\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)\)) edges contains a copy ofG. The real numberϱ (G) is defined in the same way adding the constraint that all independent sets of vertices inH have sizeo(n). Answering a problem of Erdős and Sos it is shown that there exist infinitely manyk-graphs with 0<ϱ(G)<π(G) for everyk≧3. It is worth noting that we were unable to find a singleG with the above property.

Dual vectors and lower bounds for the nearest lattice point problem

Abstract: We prove that given a point $$\overline z $$ outside a given latticeL then there is a dual vector which gives a fairly good estimate for how far from the lattice the vector is To be more precise, there is a set of translated hyperplanesH i, such thatL⊂∪ iHi andd( $$\overline z $$ ∪iHi)≧(6n 2+1)−1 d( $$\overline z $$ ,L)

Edge coloring of hypergraphs and a conjecture of ErdÖs, Faber, Lovász

Abstract: Call a bypergraphsimple if for any pairu, v of distinct vertices, there is at most one edge incident to bothu andv, and there are no edges incident to exactly one vertex. A conjecture of Erdős, Faber and Lovasz is equivalent to the statement that the edges of any simple hypergraph onn vertices can be colored with at mostn colors. We present a simple proof that the edges of a simple hypergraph onn vertices can be colored with at most [1.5n-2 colors].

Contractions of graphs with no spanning eulerian subgraphs

Abstract: Letp≧2 be a fixed integer, and letG be a connected graph onn vertices. Ifδ(G)≧2, ifd(u)+d(v)>2n/p−2 holds wheneveruv∉E(G), and ifn is sufficiently large compared top, then eitherG has a spanning eulerian subgraph, orG is contractible to a graphG1 of order less thenp and with no spanning eulerian subgraph. The casep=2 was proved by Lesniak-Foster and Williamson. The casep=5 was conjectured by Benhocine, Clark, Kohler, and Veldman, when they proved virtually the casep=3. The inequality is best-possible.

The schrijver system of odd join polyhedra

Abstract: Graphs for which the set oft-joins andt-cuts has “the max-flow-min-cut property”, i.e. for which the minimal defining system of thet-join polyhedron is totally dual integral, have been characterized by Seymour. An extension of this problem isto characterize the (uniquely existing) minimal totally dual integral defining system (Schrijver-system) of an arbitrary t-join polyhedron. This problem is solved in the present paper. The main idea is to uset-borders, a generalization oft-cuts, to obtain an integer minimax formula for the cardinality of a minimumt-join. (At-border is the set of edges joining different classes of a partition of the vertex set intot-odd sets.) It turns out that the (uniquely existing) “strongest minimax theorem” involves just this notion.

Limit theorem concerning random mapping patterns

Abstract: Mapping patterns may be represented by unlabelled directed graphs in which each point has out-degree one. Assuming uniform probability distribution on the set of all mapping patterns onn points, we obtain limit distributions of some characteristics associated with the graphs of mapping patterns (connected and disconnected), asn→∞. In particular, we study the number of points belonging to cycles, the number of cycles and components having prescribed (fixed) number of points and the total number of components.

The classification of distance-regular graphs of type IIB

Abstract: The distance-regular graphsΛ of type IIB in Bannai and Ito [1] have intersection numbers of the form Open image in new window whered is the diameter of Λ, andh, x, andt are complex constants. In this paper we show a graph of type IIB and diameterd (3≦d) is either the antipodal quotient of the Hamming graphH(2d+1,2), or has the same intersection numbers as the antipodal quotient ofH(2d, 2).

A lower bound on strictly non-blocking networks

Abstract: We prove that a strictly non-blockingn-connector of depthk must haveω(n1+1/(k−1)) edges.

Girth and residual finiteness

Abstract: On etudie la relation entre la finitude residuelle d'un groupe G et l'existence d'une suite de graphes finis dont les groupes sont les quotients de G, avec un calibre tendant vers l'infini

Balanced extensions of graphs and hypergraphs

Abstract: For a hypergraphG withv vertices ande i edges of sizei, the average vertex degree isd(G)= ∑ie 1/v. Callbalanced ifd(H)≦d(G) for all subhypergraphsH ofG. Let $$m(G) = \mathop {\max }\limits_{H \subseteqq G} d(H).$$ A hypergraphF is said to be abalanced extension ofG ifG⊂F, F is balanced andd(F)=m(G), i.e.F is balanced and does not increase the maximum average degree. It is shown that for every hypergraphG there exists a balanced extensionF ofG. Moreover everyr-uniform hypergraph has anr-uniform balanced extension. For a graphG let ext (G) denote the minimum number of vertices in any graph that is a balanced extension ofG. IfG hasn vertices, then an upper bound of the form ext(G)c 2 n 2 for an infinite family of graphs. However for sufficiently dense graphs an improved upper bound ext(G)

A short proof of the nonuniform Ray-Chaudhuri-Wilson inequality

Abstract: LetL be a set ofs nonnegative integers and ℱ a family of subsets of ann-element setX. Suppose that for any two distinct membersA,B∈ℱ we have¦A ∩ B¦∈ L. Assuming in addition that, ℱ is uniform, i.e. each member of ℱ has the same cardinality, a celebrated theorem of D. K. Ray-Chaudhuri and R. M. Wilson asserts that ¦ℱ¦≦ P. Frankl and R. M. Wilson proved that without the uniformity assumption, we have . We give a short proof of this latter result.

A remark on partial linear spaces of girth $5$ with an application to strongly regular graphs

Abstract: We derive a lower bound on the number of points of a partial linear space of girth 5. As an application, certain strongly regular graphs withμ=2 are ruled out by observing that the first subconstituents are partial linear spaces.

On a relation between a cyclic relative difference set associated with the quadratic extensions of a finite field and the Szekeres difference sets

Abstract: Letq≡ 3 (mod 4) be a prime power and put $$n = \frac{{q - 1}}{2}$$ . We consider a cyclic relative difference set with parametersq 2−1,q, 1,q−1 associated with the quadratic extension GF(q2)/GF((q). The even part and the odd part of the cyclic relative difference set taken modulon are $$2 - \left\{ {n;\frac{{n + 1}}{2};\frac{{n + 1}}{2}} \right\}$$ supplementary difference sets. Moreover it turns out that their complementary subsets are identical with the Szekeres difference sets. This result clarifies the true nature of the Szekeres difference sets. We prove these results by using the theory of the relative Gauss sums.

On a lattice point problem of L. Moser II

Abstract: In this paper we complete the proof of the following conjecture of L. Moser: Any convex region of arean can be placed on the plane so as to cover ≧n+f(n) lattice points, wheref(n) →∞.

A generalization of the ingleton—Main lemma and a class of non-algebraic matroids

Abstract: On donne de nouveaux exemples de matroides non algebriques a partir d'une generalisation du lemme de Ingleton-Main

Matrices with prescribed row, column and block sums

Abstract: The main result of the paper is Theorem 1. It concerns the sets of integral symmetric matrices with given block partition and prescribed row, column and block sums. It is shown that by interchanges preserving these sums we can pass from any two matrices, one from each set, to the other two ones falling “close” together as much as possible. One of the direct corollaries of Theorem 1 is substantiating the fact that any realization ofr-graphical integer-pair sequence can be obtained from any other one byr-switchings preserving edge degrees. This result is also of interest in connection with the problem of determinings-complete properties. In the special cases Theorem 1 includes a number of well-known results, some of which are presented.