Abstract: The main result of this paper is that every 4-uniform hypergraph on n vertices and m edges has a transversal with no more than (5n + 4m)/21 vertices. In the particular case n = m, the transversal has at most 3n/7 vertices, and this bound is sharp in the complement of the Fano plane. Chvatal and McDiarmid [5] proved that every 3-uniform hypergraph with n vertices and edges has a transversal of size n/2. Two direct corollaries of these results are that every graph with minimal degree at least 3 has total domination number at most n/2 and every graph with minimal degree at least 4 has total domination number at most 3n/7. These two bounds are sharp.

Abstract: Trevisan showed that many pseudorandom generator constructions give rise to constructions of explicit extractors. We show how to use such constructions to obtain explicit lossless condensers. A lossless condenser is a probabilistic map using only O(logn) additional random bits that maps n bits strings to poly(logK) bit strings, such that any source with support size K is mapped almost injectively to the smaller domain. Our construction remains the best lossless condenser to date. By composing our condenser with previous extractors, we obtain new, improved extractors. For small enough min-entropies our extractors can output all of the randomness with only O(logn) bits. We also obtain a new disperser that works for every entropy loss, uses an O(logn) bit seed, and has only O(logn) entropy loss. This is the best disperser construction to date, and yields other applications. Finally, our lossless condenser can be viewed as an unbalanced bipartite graph with strong expansion properties.

Abstract: We determine the maximal gap between the optimal values of an integer program and its linear programming relaxation, where the matrix and cost function are fixed but the right hand side is unspecified. Our formula involves irreducible decomposition of monomial ideals. The gap can be computed in polynomial time when the dimension is fixed.

Abstract: We introduce a natural extension of the vertex degree to ends. For the cycle space C(G) as proposed by Diestel and Kuhn [4, 5], which allows for infinite cycles, we prove that the edge set of a locally finite graph G lies in C(G) if and only if every vertex and every end has even degree. In the same way we generalise to locally finite graphs the characterisation of the cycles in a finite graph as its 2-regular connected subgraphs.

Abstract: Let Γ be a G-symmetric graph whose vertex set admits a nontrivial G-invariant partition ℬ with block size v. Let Γ ℬ be the quotient graph of Γ relative to ℬ and Γ[B,C] the bipartite subgraph of Γ induced by adjacent blocks B,C of ℬ. In this paper we study such graphs for which Γ ℬ is connected, (G, 2)-arc transitive and is almost covered by Γ in the sense that Γ[B,C] is a matching of v-1 ≥ 2 edges. Such graphs arose as a natural extremal case in a previous study by the author with Li and Praeger. The case Γ ℬ ≅K v+1 is covered by results of Gardiner and Praeger. We consider here the general case where Γ ℬ ≇K v+1, and prove that, for some even integer n ≥ 4, Γ ℬ is a near n-gonal graph with respect to a certain G-orbit on n-cycles of Γ ℬ. Moreover, we prove that every (G, 2)-arc transitive near n-gonal graph with respect to a G-orbit on n-cycles arises as a quotient Γ ℬ of a graph with these properties. (A near n-gonal graph is a connected graph Σ of girth at least 4 together with a set ℰ of n-cycles of Σ such that each 2-arc of Σ is contained in a unique member of ℰ.)

Abstract: Chudnovsky et al. gave a min-max formula for the maximum number of node-disjoint nonzero A-paths in group-labeled graphs [1], which is a generalization of Mader's theorem on node-disjoint A-paths [3]. Here we present a further generalization with a shorter proof. The main feature of Theorem 2.1 is that parity is “hidden” inside $$\ifmmode\expandafter\hat\else\expandafter\^\fi{v}$$, which is given by an oracle for non-bipartite matching.

Abstract: We say that H has an odd complete minor of order at least l if there are l vertex disjoint trees in H such that every two of them are joined by an edge, and in addition, all the vertices of trees are two-colored in such a way that the edges within the trees are bichromatic, but the edges between trees are monochromatic. Gerards and Seymour conjectured that if a graph has no odd complete minor of order l, then it is (l − 1)-colorable. This is substantially stronger than the well-known conjecture of Hadwiger. Recently, Geelen et al. proved that there exists a constant c such that any graph with no odd K k -minor is ck√logk-colorable. However, it is not known if there exists an absolute constant c such that any graph with no odd K k -minor is ck-colorable. Motivated by these facts, in this paper, we shall first prove that, for any k, there exists a constant f(k) such that every (496k + 13)-connected graph with at least f(k) vertices has either an odd complete minor of size at least k or a vertex set X of order at most 8k such that G–X is bipartite. Since any bipartite graph does not contain an odd complete minor of size at least three, the second condition is necessary. This is an analogous result of Bohme et al. We also prove that every graph G on n vertices has an odd complete minor of size at least n/2α(G) − 1, where α(G) denotes the independence number of G. This is an analogous result of Duchet and Meyniel. We obtain a better result for the case α(G)= 3.

Abstract: In [14] Matousek and Ziegler compared various topological lower bounds for the chromatic number. They proved that Lovasz’s original bound [9] can be restated as X(G) ≥ ind(B(G)) + 2. Sarkaria’s bound [15] can be formulated as X(G) ≥ ind(B0(G)) + 1. It is known that these lower bounds are close to each other, namely the difference between them is at most 1. In this paper we study these lower bounds, and the homotopy types of box complexes. The most interesting result is that up to ℤ2-homotopy the box complex B(G) can be any ℤ2-space. This together with topological constructions allows us to construct graphs showing that the mentioned two bounds are different. Some of the results were announced in [14].

Abstract: In Combinatorica 17(2), 1997, Kohayakawa, Łuczak and Rodl state a conjecture which has several implications for random graphs. If the conjecture is true, then, for example, an application of a version of Szemeredi’s regularity lemma for sparse graphs yields an estimation of the maximal number of edges in an H-free subgraph of a random graph G n, p . In fact, the conjecture may be seen as a probabilistic embedding lemma for partitions guaranteed by a version of Szemeredi’s regularity lemma for sparse graphs. In this paper we verify the conjecture for H = K 4, thereby providing a conceptually simple proof for the main result in the paper cited above.

Abstract: Combinatorial property testing, initiated by Rubinfeld and Sudan [23] and formally defined by Goldreich, Goldwasser and Ron in [18], deals with the following relaxation of decision problems: Given a fixed property P and an input f, distinguish between the case that f satisfies P, and the case that no input that differs from f in less than some fixed fraction of the places satisfies P. An (e, q)-test for P is a randomized algorithm that queries at most q places of an input f and distinguishes with probability 2/3 between the case that f has the property and the case that at least an e-fraction of the places of f need to be changed in order for it to have the property. Here we concentrate on labeled, d-dimensional grids, where the grid is viewed as a partially ordered set (poset) in the standard way (i.e. as a product order of total orders). The main result here presents an (e, poly(1/e))-test for every property of 0/1 labeled, d-dimensional grids that is characterized by a finite collection of forbidden induced posets. Such properties include the “monotonicity” property studied in [9,8,13], other more complicated forbidden chain patterns, and general forbidden poset patterns. We also present a (less efficient) test for such properties of labeled grids with larger fixed size alphabets. All the above tests have in addition a 1-sided error probability. This class of properties is related to properties that are defined by certain first order formulae with no quantifier alternation over the syntax containing the grid order relations. We also show that with one quantifier alternation, a certain property can be defined, for which no test with query complexity of O(n 1/4) (for a small enough fixed e) exists. The above results identify new classes of properties that are defined by means of restricted logics, and that are efficiently testable. They also lay out a platform that bridges some previous results.

Abstract: A colored graph is a complete graph in which a color has been assigned to each edge, and a colorful cycle is a cycle in which each edge has a different color. We first show that a colored graph lacks colorful cycles iff it is Gallai, i.e., lacks colorful triangles. We then show that, under the operation mon ≡ m + n − 2, the omitted lengths of colorful cycles in a colored graph form a monoid isomorphic to a submonoid of the natural numbers which contains all integers past some point. We prove that several but not all such monoids are realized. We then characterize exact Gallai graphs, i.e., graphs in which every triangle has edges of exactly two colors. We show that these are precisely the graphs which can be iteratively built up from three simple colored graphs, having 2, 4, and 5 vertices, respectively. We then characterize in two different ways the monochromes, i.e., the connected components of maximal monochromatic subgraphs, of exact Gallai graphs. The first characterization is in terms of their reduced form, a notion which hinges on the important idea of a full homomorphism. The second characterization is by means of a homomorphism duality.

Abstract: In 1963, Tibor Gallai [9] asked whether every strongly connected directed graph D is spanned by α directed circuits, where α is the stability of D. We give a proof of this conjecture.

Abstract: Any countably infinite tournament T0 embeds as a moiety of the random tournament T in such a way that its setwise stabilizer in Aut(T) is isomorphic to Aut(T0).

Abstract: Solving a problem of Diestel [9] relevant to the theory of cycle spaces of infinite graphs, we show that the Freudenthal compactification of a locally finite graph can have connected subsets that are not path-connected. However we prove that connectedness and path-connectedness to coincide for all but a few sets, which have a complicated structure.

Abstract: A maximal antichain A of poset P splits if and only if there is a set B ⊂ A such that for each p ∈ P either b ≤ p for some b ∈ B or p ≤ c for some c ∈ A\B. The poset P is cut-free if and only if there are no x < y < z in P such that [x,z]P = [x,y]P ∪ [y,z]P . By [1] every maximal antichain in a finite cut-free poset splits. Although this statement for infinite posets fails (see [2])) we prove here that if a maximal antichain in a cut-free poset “resembles” to a finite set then it splits. We also show that a version of this theorem is just equivalent to Axiom of Choice. We also investigate possible strengthening of the statements that “A does not split” and we could find a maximal strengthening.

Abstract: What is the maximum possible number, f3(n), of vectors of length n over {0,1,2} such that the Hamming distance between every two is even? What is the maximum possible number, g3(n), of vectors in {0,1,2}n such that the Hamming distance between every two is odd? We investigate these questions, and more general ones, by studying Xor powers of graphs, focusing on their independence number and clique number, and by introducing two new parameters of a graph G. Both parameters denote limits of series of either clique numbers or independence numbers of the Xor powers of G (normalized appropriately), and while both limits exist, one of the series grows exponentially as the power tends to infinity, while the other grows linearly. As a special case, it follows that f3(n) = Θ(2n) whereas g3(n)=Θ(n).

Abstract: In this paper we present a Berge-Tutte-type theorem for a matching problem in directed graphs. This extends the maximum matching problem in undirected graphs, the maximum even factor problem in weakly symmetric directed graphs proposed by W. H. Cunningham and J. F. Geelen in [6], and a packing problem for cycles and edges in undirected graphs. We show an Edmonds-Gallai-type structural description of a canonical set attaining the minimum in the formula. We also give a generalization of the matching matroid to this concept.

Abstract: In the cake cutting problem, n≥2 players want to cut a cake into n pieces so that every player gets a ‘fair’ share of the cake by his own measure. We prove the following result: For every e>0, there exists a cake division scheme for n players that uses at most cen cuts, and in which each player can enforce to get a share of at least (1-e)/n of the cake according to his own private measure.

Abstract: Suppose that C is a balanced simplicial complex. We show that its flag f-vector contains an interesting multiplicative structure. We define η s (C):= log2 f S (C), and characterize the convex cone in which this flag η-vector may lie. Additionally, we specialize our results to the case when C is a pure balanced simplicial complex, and when C is a graded poset.

Abstract: The k-th power of a graph G is the graph whose vertex set is V(G) k , where two distinct k-tuples are adjacent iff they are equal or adjacent in G in each coordinate. The Shannon capacity of G, c(G), is lim k→∞ α(G k )1/k , where α(G) denotes the independence number of G. When G is the characteristic graph of a channel C, c(G) measures the effective alphabet size of C in a zero-error protocol. A sum of channels, C = Σ i C i , describes a setting when there are t ≥ 2 senders, each with his own channel C i , and each letter in a word can be selected from any of the channels. This corresponds to a disjoint union of the characteristic graphs, G = Σ i G i . It is well known that c(G) ≥ Σ i c(G i ), and in [1] it is shown that in fact c(G) can be larger than any fixed power of the above sum. We extend the ideas of [1] and show that for every F, a family of subsets of [t], it is possible to assign a channel C i to each sender i ∈ [t], such that the capacity of a group of senders X ⊂ [t] is high iff X contains some F ∈ F. This corresponds to a case where only privileged subsets of senders are allowed to transmit in a high rate. For instance, as an analogue to secret sharing, it is possible to ensure that whenever at least k senders combine their channels, they obtain a high capacity, however every group of k − 1 senders has a low capacity (and yet is not totally denied of service). In the process, we obtain an explicit Ramsey construction of an edge-coloring of the complete graph on n vertices by t colors, where every induced subgraph on exp $$(\Omega (\sqrt {\log n\log \log n} ))$$ vertices contains all t colors.

Abstract: In the radio channel assignment problems considered here, we must assign a ‘channel’ from the set 1,2,... of positive integers to each of n transmitters, and we wish to minimise the span of channels used, subject to the assignment leading to an acceptable level of interference. A standard form of this problem is the ‘constraint matrix’ model. The simplest case of this model (the 0, 1 case) is essentially graph colouring. We consider here a random model for the next simplest case (with lengths 0, 1 or 2), and determine the asymptotic behaviour of the span of channels needed as n→∞. We find that there is a ‘phase change’ in this behaviour, depending on the probabilities for the different lengths.

Abstract: A complete partition of a graph G is a partition of its vertex set in which any two distinct classes are connected by an edge. Let cp(G) denote the maximum number of classes in a complete partition of G. This measure was defined in 1969 by Gupta [19], and is known to be NP-hard to compute for several classes of graphs. We obtain essentially tight lower and upper bounds on the approximability of this problem. We show that there is a randomized polynomial-time algorithm that given a graph G with n vertices, produces a complete partition of size Ω(cp(G)/√lgn). This algorithm can be derandomized. We show that the upper bound is essentially tight: there is a constant C > 1, such that if there is a randomized polynomial-time algorithm that for all large n, when given a graph G with n vertices produces a complete partition into at least C·cp(G)/√lgn classes, then NP ⊆ RTime(n O(lg lg n)). The problem of finding a complete partition of a graph is thus the first natural problem whose approximation threshold has been determined to be of the form Θ((lgn) c ) for some constant c strictly between 0 and 1.

Abstract: We raise the following problem. For natural numbers m, n ≥ 2, determine pairs d′, d″ (both depending on m and n only) with the property that in every pair of set systems A, B with |A| ≤ m, |B| ≤ n, and A ∩ B ≠ 0 for all A ∈ A, B ∈ B, there exists an element contained in at least d′ |A| members of A and d″ |B| members of B. Generalizing a previous result of Kyureghyan, we prove that all the extremal pairs of (d′, d″) lie on or above the line (n − 1) x + (m − 1) y = 1. Constructions show that the pair (1 + ɛ / 2n − 2, 1 + ɛ / 2m − 2) is infeasible in general, for all m, n ≥ 2 and all ɛ > 0. Moreover, for m = 2, the pair (d′, d″) = (1 / n, 1 / 2) is feasible if and only if 2 ≤ n ≤ 4. The problem originates from Razborov and Vereshchagin’s work on decision tree complexity.

Abstract: Let Π be a projective plane of order n in Lenz–Barlotti class I.4, and assume that n is a multiple of 3. Then either n=3 or n is a multiple of 9.

Abstract: Let H n be the hypercube {0, 1} n , and denote by H n,p Bernoulli bond percolation on H n , with parameter p = n −α . It is shown that at α = 1/2 there is a phase transition for the metric distortion between H n and H n,p . For α < 1/2, the giant component of H n,p is likely to be quasi-isometric to H n with constant distortion (depending only on α). For 1/2 < α < 1 the minimal distortion tends to infinity as a power of n. We argue that the phase 1/2 < α < 1 is an analogue of the non-uniqueness phase appearing in percolation on non-amenable graphs.

Abstract: We resolve the following conjecture raised by Levin together with Linial, London, and Rabinovich [Combinatorica, 1995]. For a graph G, let dim(G) be the smallest d such that G occurs as a (not necessarily induced) subgraph of ℤ∞ d , the infinite graph with vertex set ℤ d and an edge (u, v) whenever ∥u − v∥∞ = 1. The growth rate of G, denoted ρ G , is the minimum ρ such that every ball of radius r > 1 in G contains at most r ρ vertices. By simple volume arguments, dim(G) = Ω(ρ G ). Levin conjectured that this lower bound is tight, i.e., that dim(G) = O(ρ G ) for every graph G. Previously, it was unknown whether dim(G) could be bounded above by any function of ρ G . We show that a weaker form of Levin’s conjecture holds by proving that dim(G) = O(ρ G log ρ G ) for any graph G. We disprove, however, the specific bound of the conjecture and show that our upper bound is tight by exhibiting graphs for which dim(G) = Ω(ρ G log ρ G ). For several special families of graphs (e.g., planar graphs), we salvage the strong form, showing that dim(G) = O(ρ G ). Our results extend to a variant of the conjecture for finite-dimensional Euclidean spaces posed by Linial and independently by Benjamini and Schramm.

Abstract: A sufficient condition for graphs with circular flow index less than 4 is found in this paper. In particular, we give a simple proof of a result obtained by Galluccio and Goddyn (Combinatorica, 2002), and obtain a larger family of such graphs.

Abstract: Gluck has proven that triangulated 2-spheres are generically 3-rigid. Equivalently, planar graphs are generically 3-stress free. We show that already the K 5-minor freeness guarantees the stress freeness. More generally, we prove that every K r+2-minor free graph is generically r-stress free for 1≤r≤4. (This assertion is false for r≥6.) Some further extensions are discussed.

Abstract: The following conjecture may have never been explicitly stated, but seems to have been floating around: if the vertex set of a graph with maximal degree Δ is partitioned into sets V i of size 2Δ, then there exists a coloring of the graph by 2Δ colors, where each color class meets each V i at precisely one vertex. We shall name it the strong 2Δ-colorability conjecture. We prove a fractional version of this conjecture. For this purpose, we prove a weighted generalization of a theorem of Haxell, on independent systems of representatives (ISR’s). En route, we give a survey of some recent developments in the theory of ISR’s.