Two Questions of Erdős on Hypergraphs above the Turán Threshold

For ordinary graphs it is known that any graph G with more edges than the Turán number of must contain several copies of , and a copy of , the complete graph on vertices with one missing edge. Erd?s asked if the same result is true for , the complete 3‐uniform hypergraph on s vertices. In this note, we show that for small values of n, the number of vertices in G, the answer is negative for . For the second property, that of containing a , we show that for the answer is negative for all large n as well, by proving that the Turán density of is greater than that of .     

Extending the Erdős–Ko–Rado theorem

Let be a k‐uniform hypergraph on n vertices. Suppose that holds for all . We prove that the size of is at most if satisfies and n is sufficiently large. © 2005 Wiley Periodicals, Inc. J Combin Designs     

Strengthening Theorems of Dirac and Erdős on Disjoint Cycles

Let be an integer, be the set of vertices of degree at least 2k in a graph G , and be the set of vertices of degree at most in G . In 1963, Dirac and Erd?s proved that G contains k (vertex) disjoint cycles whenever . The main result of this article is that for , every graph G with containing at most t disjoint triangles and with contains k disjoint cycles. This yields that if and , then G contains k disjoint cycles. This generalizes the Corrádi–Hajnal Theorem, which states that every graph G with and contains k disjoint cycles.     

The Erdős–Nagy theorem and its ramifications

Given a simple polygon in the plane, a flip is defined as follows: consider the convex hull of the polygon. If there are no pockets do not perform a flip. If there are pockets then reflect one pocket across its line of support of the polygon to obtain a new simple polygon. In 1934 Paul Erdős introduced the problem of repeatedly flipping all the pockets of a simple polygon simultaneously and he conjectured that the polygon would become convex after a finite number of flips. In 1939 Béla Nagy proved that if at each step only one pocket is flipped the polygon will become convex after a finite number of flips. The history of this problem is reviewed, and a simple elementary proof is given of a stronger version of the theorem. Variants, generalizations, and applications of the theorem of interest in computational knot theory, polymer physics and molecular biology are discussed. Several results in the literature are improved with the application of the theorem. For example, Grünbaum and Zaks recently showed that even non-simple (self-crossing) polygons may be convexified in a finite number of suitable flips. Their flips each take Θ(n²) time to determine. A simpler proof of this result is given that yields an algorithm that takes O(n) time to determine each flip. In the context of knot theory Millet proposed an algorithm for convexifying equilateral polygons in 3-dimensions with a generalization of a flip called a pivot. Here Millet's algorithm is generalized so that it works also in dimensions higher than three and for polygons containing edges with arbitrary lengths. A list of open problems is included.     

An extension of the Erdős–Ginzburg–Ziv Theorem to hypergraphs

An n-set partition of a sequence S is a collection of n nonempty subsequences of S, pairwise disjoint as sequences, such that every term of S belongs to exactly one of the subsequences, and the terms in each subsequence are all distinct with the result that they can be considered as sets. For a sequence S, subsequence S^′, and set T, |T∩S| denotes the number of terms x of S with xT, and |S| denotes the length of S, and SS^′ denotes the subsequence of S obtained by deleting all terms in S^′. We first prove the following two additive number theory results.(1) Let S be a finite sequence of elements from an abelian group G. If S has an n-set partition, A=A₁,…,A_n, such that

then there exists a subsequence S^′ of S, with length |S^′|≤max{|S|−n+1,2n}, and with an n-set partition, , such that . Furthermore, if ||A_i|−|A_j||≤1 for all i and j, or if |A_i|≥3 for all i, then .(2) Let S be a sequence of elements from a finite abelian group G of order m, and suppose there exist a,bG such that . If |S|≥2m−1, then there exists an m-term zero-sum subsequence S^′ of S with or .Let be a connected, finite m-uniform hypergraph, and be the least integer n such that for every 2-coloring (coloring with the elements of the cyclic group ) of the vertices of the complete m-uniform hypergraph , there exists a subhypergraph isomorphic to such that every edge in is monochromatic (such that for every edge e in the sum of the colors on e is zero). As a corollary to the above theorems, we show that if every subhypergraph of contains an edge with at least half of its vertices monovalent in , or if consists of two intersecting edges, then . This extends the Erdős–Ginzburg–Ziv Theorem, which is the case when is a single edge.     

Erdős–Ko–Rado theorems for permutations and set partitions

Let Sym([n]) denote the collection of all permutations of [n]={1,…,n}. Suppose is a family of permutations such that any two of its elements (when written in its cycle decomposition) have at least t cycles in common. We prove that for sufficiently large n, with equality if and only if is the stabilizer of t fixed points. Similarly, let denote the collection of all set partitions of [n] and suppose is a family of set partitions such that any two of its elements have at least t blocks in common. It is proved that, for sufficiently large n, with equality if and only if consists of all set partitions with t fixed singletons, where B_n is the nth Bell number.     

On a Conjecture of Erdős,Gallai, and Tuza

Erd?s, Gallai, and Tuza posed the following problem: given an n‐vertex graph G, let denote the smallest size of a set of edges whose deletion makes G triangle‐free, and let denote the largest size of a set of edges containing at most one edge from each triangle of G. Is it always the case that ? We have two main results. We first obtain the upper bound , as a partial result toward the Erd?s–Gallai–Tuza conjecture. We also show that always , where m is the number of edges in G; this bound is sharp in several notable cases.     

On a generalization of a theorem of Erdős and Fuchs

Let A={a₁,a₂,…}(a₁<a₂<) be an infinite sequence of nonnegative integers, let k≥2 be a fixed integer and denote by r_k(A,n) the number of solutions of a_i1+a_i2++a_ik≤n. Montgomery and Vaughan proved that r₂(A,n)=cn+o(n^1/4) cannot hold for any constant c>0. In this paper, we extend this result to k>2.     

On a problem of Duke–Erdős–Rödl on cycle-connected subgraphs

In this short note, we prove that for β<1/5 every graph G with n vertices and n^2−β edges contains a subgraph G^′ with at least cn^2−2β edges such that every pair of edges in G^′ lie together on a cycle of length at most 8. Moreover edges in G^′ which share a vertex lie together on a cycle of length at most 6. This result is best possible up to the constant factor and settles a conjecture of Duke, Erdős, and Rödl.     

The Erdős–Pósa Property for Long Circuits

For an integer ? at least 3, we prove that if G is a graph containing no two vertex‐disjoint circuits of length at least ?, then there is a set X of at most vertices that intersects all circuits of length at least ?. Our result improves the bound due to Birmelé, Bondy, and Reed (The Erd?s–Pósa property for long circuits, Combinatorica 27 (2007), 135–145) who conjecture that ? vertices always suffice.     

A Relaxed Version of the Erdős–Lovász Tihany Conjecture

The Erd?s–Lovász Tihany conjecture asserts that every graph G with ) contains two vertex disjoint subgraphs G ₁ and G ₂ such that and . Under the same assumption on G , we show that there are two vertex disjoint subgraphs G ₁ and G ₂ of G such that (a) and or (b) and . Here, is the chromatic number of is the clique number of G , and col(G ) is the coloring number of G .     

Generalization of a theorem of Erdős and Rényi on Sidon sequences

Erd?s and Rényi claimed and Vu proved that for all h ≥ 2 and for all ? > 0, there exists g = g_h(?) and a sequence of integers A such that the number of ordered representations of any number as a sum of h elements of A is bounded by g, and such that |A ∩ [1,x]| ? x^1/h‐?. We give two new proofs of this result. The first one consists of an explicit construction of such a sequence. The second one is probabilistic and shows the existence of such a g that satisfies g_h(?) ? ?^?1, improving the bound g_h(?) ? ?^?h+1 obtained by Vu. Finally we use the “alteration method” to get a better bound for g₃(?), obtaining a more precise estimate for the growth of B₃[g] sequences. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Random coloring method in the combinatorial problem of Erdős and Lovász

The work deals with a combinatorial problem of P. Erd?s and L. Lovász concerning simple hypergraphs. Let denote the minimum number of edges in an n‐uniform simple hypergraph with chromatic number at least . The main result of the work is a new asymptotic lower bound for . We prove that for large n and r satisfying the following inequality holds where . This bound improves previously known bounds for . The proof is based on a method of random coloring. We have also obtained results concerning colorings of h‐simple hypergraphs. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

The Erdős‐Sós Conjecture for trees of diameter four

The Erd?s‐Sós Conjecture is that a finite graph G with average degree greater than k ? 2 contains every tree with k vertices. Theorem 1 is a special case: every k‐vertex tree of diameter four can be embedded in G. A more technical result, Theorem 2, is obtained by extending the main ideas in the proof of Theorem 1. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 291–301, 2005     

An extension of the Erdős‐Tetali theorem

Given a sequence , let r_??,h(n) denote the number of ways n can be written as the sum of h elements of ??. Fixing h ≥ 2, we show that if f is a suitable real function (namely: locally integrable, O‐regularly varying and of positive increase) satisfying then there must exist with for which r_{??,h + ?}(n) = Θ(f(n)^{h + ?}/n) for all ? ≥ 0. Furthermore, for h = 2 this condition can be weakened to . The proof is somewhat technical and the methods rely on ideas from regular variation theory, which are presented in an appendix with a view towards the general theory of additive bases. We also mention an application of these ideas to Schnirelmann's method.     

On a problem of Erdős on integers, none of which divides the product of others

Erdős estimated the maximal number of integers selected from {1,2,…,N}, so that none of them divides the product of two others. In this paper, Erdős’ problem is extended to sets of integers such that none of them divides the product of k others. The proofs use combinatorial results.     

Some remarks on the Lehmer conjecture

In 1933, Lehmer exhibited the polynomial

$$\begin{aligned} L(z)=z^{10} + z^9 - z^7 - z^6 - z^5 - z^4 - z^3 + z + 1 \end{aligned}$$

with Mahler measure \(\mu _0>1\). Then he asked if \(\mu _0\) is the smallest Mahler measure, not 1. This question became known as the Lehmer conjecture and it was apparently solved in the positive, while this paper was in preparation [19]. In this paper we consider those polynomials of the form \(\chi _A\), that is, Coxeter polynomials of a finite dimensional algebra A (for instance \(L(z)=\chi _{\mathbb {E}_{10}}\)). A polynomial in \(\mathbb {Z}[T]\) which is either cyclotomic or with Mahler measure \(\ge \mu _0\) is called a Lehmer polynomial. We give some necessary conditions for a polynomial to be Lehmer. We show that A being a tree algebra is a sufficient condition for \(\chi _A\) to be Lehmer.

     

Modularity of Erdős‐Rényi random graphs

For a given graph G, each partition of the vertices has a modularity score, with higher values indicating that the partition better captures community structure in G. The modularity q^?(G) of the graph G is defined to be the maximum over all vertex partitions of the modularity score, and satisfies 0 ≤ q^?(G)<1. Modularity is at the heart of the most popular algorithms for community detection. We investigate the behaviour of the modularity of the Erd?s‐Rényi random graph G_n,p with n vertices and edge‐probability p. Two key findings are that the modularity is 1+o(1) with high probability (whp) for np up to 1+o(1) and no further; and when np ≥ 1 and p is bounded below 1, it has order (np)^?1/2 whp, in accord with a conjecture by Reichardt and Bornholdt in 2006. We also show that the modularity of a graph is robust to changes in a few edges, in contrast to the sensitivity of optimal vertex partitions.     

Quadratic Upper Bounds on the Erdős–Pósa Property for a Generalization of Packing and Covering Cycles

According to the classical Erd?s–Pósa theorem, given a positive integer k, every graph G either contains k vertex disjoint cycles or a set of at most vertices that hits all its cycles. Robertson and Seymour (J Comb Theory Ser B 41 (1986), 92–114) generalized this result in the best possible way. More specifically, they showed that if is the class of all graphs that can be contracted to a fixed planar graph H, then every graph G either contains a set of k vertex‐disjoint subgraphs of G, such that each of these subgraphs is isomorphic to some graph in or there exists a set S of at most vertices such that contains no subgraph isomorphic to any graph in . However, the function f is exponential. In this note, we prove that this function becomes quadratic when consists all graphs that can be contracted to a fixed planar graph . For a fixed c, is the graph with two vertices and parallel edges. Observe that for this corresponds to the classical Erd?s–Pósa theorem.     

On the Loebl–Komlós–Sós conjecture

The Loebl–Komlós–Sós conjecture says that any graph G on n vertices with at least half of vertices of degree at least k contains each tree of size k. We prove that the conjecture is true for paths as well as for large values of k(k ≥ n − 3). © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 269–276, 2000