Note making a <Emphasis Type="Italic">K</Emphasis><Subscript>4</Subscript>-free graph bipartite

We show that every K ₄-free graph G with n vertices can be made bipartite by deleting at most n ²/9 edges. Moreover, the only extremal graph which requires deletion of that many edges is a complete 3-partite graph with parts of size n/3. This proves an old conjecture of P. Erdős. Research supported in part by NSF CAREER award DMS-0546523, NSF grant DMS-0355497, USA-Israeli BSF grant, and by an Alfred P. Sloan fellowship.     

Triangle Factors In Sparse Pseudo-Random Graphs

The goal of the paper is to initiate research towards a general, Blow-up Lemma type embedding statement for pseudo-random graphs with sublinear degrees. In particular, we show that if the second eigenvalue of a d-regular graph G on 3n vertices is at most cd ³/n ² log n, for some sufficiently small constant c > 0, then G contains a triangle factor. We also show that a fractional triangle factor already exists if < 0.1d ²/n. The latter result is seen to be best possible up to a constant factor, for various values of the degree d = d(n).* Supported by a USA-Israeli BSF grant, by a grant from the Israel Science Foundation and by a Bergmann Memorial Award. Research supported in part by NSF grants DMS-0106589, CCR-9987845 and by the State of New Jersey. Research supported in part by NSF grant DMS 99-70270 and by the joint Berlin/Zurich graduate program Combinatorics, Geometry, Computation, financed by the German Science Foundation (DFG) and ETH Zürich.     

An approximate Dirac-type theorem for k-uniform hypergraphs

A k-uniform hypergraph is hamiltonian if for some cyclic ordering of its vertex set, every k consecutive vertices form an edge. In 1952 Dirac proved that if the minimum degree in an n-vertex graph is at least n/2 then the graph is hamiltonian. We prove an approximate version of an analogous result for uniform hypergraphs: For every K ≥ 3 and γ > 0, and for all n large enough, a sufficient condition for an n-vertex k-uniform hypergraph to be hamiltonian is that each (k − 1)-element set of vertices is contained in at least (1/2 + γ)n edges. Research supported by NSF grant DMS-0300529. Research supported by KBN grant 2P03A 015 23 and N201036 32/2546. Part of research performed at Emory University, Atlanta. Research supported by NSF grant DMS-0100784.     

Generating Random Regular Graphs

Random regular graphs play a central role in combinatorics and theoretical computer science. In this paper, we analyze a simple algorithm introduced by Steger and Wormald [10] and prove that it produces an asymptotically uniform random regular graph in a polynomial time. Precisely, for fixed d and n with d = O(n^1/3−ε), it is shown that the algorithm generates an asymptotically uniform random d-regular graph on n vertices in time O(nd²). This confirms a conjecture of Wormald. The key ingredient in the proof is a recently developed concentration inequality by the second author. The algorithm works for relatively large d in practical (quadratic) time and can be used to derive many properties of uniform random regular graphs. * Research supported in part by grant RB091G-VU from UCSD, by NSF grant DMS-0200357 and by an A. Sloan fellowship.     

A New Lower Bound For A Ramsey-Type Problem

Let 3 ≤ r < s be fixed integers and let G be a graph on n vertices not containing a complete graph on s vertices. The main aim of this paper is to provide a new lower bound on the size of the maximum subset of G without a copy of complete graph K_r. Our results substantially improve previous bounds of Krivelevich and Bollobás and Hind. * Research supported in part by NSF grants DMS-0106589, CCR-9987845 and by the State of New Jersey. Part of this research was done while visiting Microsoft Research.     

Vertex Covers by Edge Disjoint Cliques

Dedicated to the memory of Paul Erdős Let H be a simple graph having no isolated vertices. An (H,k)-vertex-cover of a simple graph G = (V,E) is a collection of subgraphs of G satisfying 1.  , for all i = 1, ..., r, 2.  , 3.  , for all , and 4.  each is in at most k of the . We consider the existence of such vertex covers when H is a complete graph, , in the context of extremal and random graphs. Received October 31, 1999 RID="*" ID="*" Supported in part by NSF grant DMS-9627408. RID="†" ID="†" Supported in part by NSF grant CCR-9530974. RID="‡" ID="‡" Supported in part by OTKA Grants T 030059 and T 29074, FKFP 0607/1999 and by the Bolyai Foundation. RID="§" ID="§" Supported in part by NSF grant DMS-9970622.     

On The Structure Of Triangle-Free Graphs Of Large Minimum Degree

It is shown that for every ε>0 there exists a constant L such that every triangle-free graph on n vertices with minimum degree at least (1/3+ε)n is homomorphic to a triangle-free graph on at most L vertices. * Research partially supported by KBN grant 2 P03A 016 23.     

Density theorems for bipartite graphs and related Ramsey-type results

In this paper, we present several density-type theorems which show how to find a copy of a sparse bipartite graph in a graph of positive density. Our results imply several new bounds for classical problems in graph Ramsey theory and improve and generalize earlier results of various researchers. The proofs combine probabilistic arguments with some combinatorial ideas. In addition, these techniques can be used to study properties of graphs with a forbidden induced subgraph, edge intersection patterns in topological graphs, and to obtain several other Ramsey-type statements. Research supported by an NSF Graduate Research Fellowship and a Princeton Centennial Fellowship. Research supported in part by NSF CAREER award DMS-0812005 and by USA-Israeli BSF grant.     

A parallel algorithm for the maximal path problem

In this paper we present anO (log⁵ n) time parallel algorithm for constructing a Maximal Path in an undirected graph. We also give anO (log^1/2+ε) time parallel algorithm for constructing a depth first search tree in an undirected graph. This work was supported in part by an IBM Faculty Development Award, an NSF Graduate Fellowship, and NSF grant DCR-8351757.     

On a graph packing conjecture by Bollobás,Eldridge and Catlin

Two graphs G ₁ and G ₂ of order n pack if there exist injective mappings of their vertex sets into [n], such that the images of the edge sets are disjoint. In 1978, Bollobás and Eldridge, and independently Catlin, conjectured that if (Δ(G ₁) + 1)(Δ(G ₂) + 1) ≤ n + 1, then G ₁ and G ₂ pack. Towards this conjecture, we show that for Δ(G ₁),Δ(G ₂) ≥ 300, if (Δ(G ₁) + 1)(Δ(G ₂) + 1) ≤ 0.6n + 1, then G ₁ and G ₂ pack. This is also an improvement, for large maximum degrees, over the classical result by Sauer and Spencer that G ₁ and G ₂ pack if Δ(G ₁)Δ(G ₂) < 0.5n. This work was supported in part by NSF grant DMS-0400498. The work of the second author was also partly supported by NSF grant DMS-0650784 and grant 05-01-00816 of the Russian Foundation for Basic Research. The work of the third author was supported in part by NSF grant DMS-0652306.     

Ramsey-type Theorems with Forbidden Subgraphs

Dedicated to the memory of Paul Erdős   A graph is called H-free if it contains no induced copy of H. We discuss the following question raised by Erdős and Hajnal. Is it true that for every graph H, there exists an such that any H-free graph with n vertices contains either a complete or an empty subgraph of size at least ? We answer this question in the affirmative for a special class of graphs, and give an equivalent reformulation for tournaments. In order to prove the equivalence, we establish several Ramsey type results for tournaments. Received August 22, 1999 RID="*" ID="*" Supported by a USA Israeli BSF grant, by a grant from the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. RID="†" ID="†" Supported by NSF grant CR-9732101, PSC-CUNY Research Award 663472, and OTKA-T-020914. RID="‡" ID="‡" Supported by TKI grant Stochastics@TUB, and OTKA-T-026203.     

Packing directed circuits

We prove a conjecture of Younger, that for every integern0 there exists an integert0 such that for every digraphG, eitherG hasn vertex-disjoint directed circuits, orG can be made acyclic by deleting at mostt vertices.Research partially supported by DONET ECHM contract CHRXCT930090.Research partially supported by DIMACS, by NSF grant DMS-9401981 and by ONR grant N00014-92-J-1965, and partially performed under a consulting agreement with Bellcore.Research partially supported by DIMACS, by Université de Paris VI, by NSF grant DMS-9303761 and by ONR grant N00014-93-1-0325, and partially performed under a consulting agreement with Bellcore.     

How Many Ways Can One Draw A Graph?

Using results from extremal graph theory, we determine the asymptotic number of string graphs with n vertices, i.e., graphs that can be obtained as the intersection graph of a system of continuous arcs in the plane. The number becomes much smaller, for any fixed d, if we restrict our attention to systems of arcs, any two of which cross at most d times. As an application, we estimate the number of different drawings of the complete graph K_n with n vertices under various side conditions. Dedicated to Miklós Simonovits on his sixtieth birthday * Supported by NSF grant CR-00-98246, PSC-CUNY Research Award 62450-0031 and OTKA-T-032452. † Supported by OTKA-T-032452 and OTKA-T-038397.     

The Number Of Orientations Having No Fixed Tournament

Let T be a fixed tournament on k vertices. Let D(n,T ) denote the maximum number of orientations of an n-vertex graph that have no copy of T. We prove that for all sufficiently (very) large n, where t_k−1(n) is the maximum possible number of edges of a graphon n vertices with no K_k, (determined by Turán’s Theorem). The proof is based on a directed version of Szemerédi’s regularity lemma together with some additional ideas and tools from Extremal Graph Theory, and provides an example of a precise result proved by applying this lemma. For the two possible tournaments with three vertices we obtain separate proofs that avoid the use of the regularity lemma and therefore show that in these cases already holds for (relatively) small values of n. * Research supported in part by a USA Israeli BSF grant, by a grant from the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University.     

On Bipartite Graphs with Linear Ramsey Numbers

Dedicated to the memory of Paul Erdős We provide an elementary proof of the fact that the ramsey number of every bipartite graph H with maximum degree at most is less than . This improves an old upper bound on the ramsey number of the n-cube due to Beck, and brings us closer toward the bound conjectured by Burr and Erdős. Applying the probabilistic method we also show that for all and there exists a bipartite graph with n vertices and maximum degree at most whose ramsey number is greater than for some absolute constant c>1. Received December 1, 1999 RID="*" ID="*" Supported by NSF grant DMS-9704114 RID="**" ID="**" Supported by KBN grant 2 P03A 032 16     

Bounds on the convex label number of trees

A convex labelling of a tree is an assignment of distinct non-negative integer labels to vertices such that wheneverx, y andz are the labels of vertices on a path of length 2 theny≦(x+z)/2. In addition if the tree is rooted, a convex labelling must assign 0 to the root. The convex label number of a treeT is the smallest integerm such thatT has a convex labelling with no label greater thanm. We prove that every rooted tree (and hence every tree) withn vertices has convex label number less than 4n. We also exhibitn-vertex trees with convex label number 4n/3+o(n), andn-vertex rooted trees with convex label number 2n +o(n). The research by M. B. and A. W. was partly supported by NSF grant MCS—8311422.     

Codes And Xor Graph Products

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). * Research supported in part by a USA-Israeli BSF grant, by the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. † Research partially supported by a Charles Clore Foundation Fellowship.     

On 2-Detour Subgraphs of the Hypercube

A spanning subgraph H of a graph G is a 2-detour subgraph of G if for each x, y ∈ V(G), d _H (x, y) ≤ d _G (x, y) + 2. We prove a conjecture of Erdős, Hamburger, Pippert, and Weakley by showing that for some positive constant c and every n, each 2-detour subgraph of the n-dimensional hypercube Q _n has at least clog₂ n · 2 ⁿ edges. József Balogh: Research supported in part by NSF grants DMS-0302804, DMS-0603769 and DMS-0600303, UIUC Campus Reseach Board #06139 and #07048, and OTKA 049398. Alexandr Kostochka: Research supported in part by NSF grants DMS-0400498 and DMS-0650784, and grant 06-01-00694 of the Russian Foundation for Basic Research.     

On the support of harmonic measure for the random walk

We show that harmonic measure for the simple random walk on then ×…×n cube in thed-dimensional lattice is supported on o(n ^d ) vertices. This research was supported in part by NSF grant # DMS-9353149.     

A note on a spanning 3-tree

A tree T is called a k-tree, if the maximum degree of T is at most k. In this paper, we prove that if G is an n-connected graph with independence number at most n + m + 1 (n≥1,n≥m≥0), then G has a spanning 3-tree T with at most m vertices of degree 3.