Quasi-random tournaments

2010

Few families of tournaments satisfying the n-e.c. adjacency property are known. We supply a new random construction for generating infinite families of vertex-transitive n-e.c. tournaments by considering circulant tournaments. Switching is used to generate new n-e.c. tournaments of certain orders. With aid of a computer search, we demonstrate that there is a unique minimum order 3-e.c. tournament of order 19, and there are no 3-e.c. tournaments of orders 20, 21, and 22. We show that there are no 4-e.c. tournaments of orders 47 and 48 improving the lower bound for the minimum order of such a tournament.

Journal of Graph Theory, 1998

We survey results concerning various generalizations of tournaments. The reader will see that tournaments are by no means the only class of directed graphs with a very rich structure. We describe, among numerous other topics mostly related to paths and cycles, results on hamiltonian paths and cycles. The reader will see that although these problems are polynomially solvable for all of the classes described, they can be highly non-trivial, even for these "tournament-like" digraphs.

Combinatorics, Probability and Computing, 2015

In the tournament game two players, called Maker and Breaker, alternately take turns in claiming an unclaimed edge of the complete graph K n and selecting one of the two possible orientations. Before the game starts, Breaker fixes an arbitrary tournament T k on k vertices. Maker wins if, at the end of the game, her digraph contains a copy of T k ; otherwise Breaker wins. In our main result, we show that Maker has a winning strategy for k = (2 − o(1)) log 2 n, improving the constant factor in previous results of Beck and the second author. This is asymptotically tight since it is known that for k = (2 − o(1)) log 2 n Breaker can prevent that the underlying graph of Maker's graph contains a k-clique. Moreover the precise value of our lower bound differs from the upper bound only by an additive constant of 12.

Journal of Combinatorial Theory, Series A, 2002

Haviland and Thomason and Chung and Graham were the first to investigate systematically some properties of quasi-random hypergraphs. In particular, in a series of articles, Chung and Graham considered several quite disparate properties of random-like hypergraphs of density 1/2 and proved that they are in fact equivalent. The central concept in their work turned out to be the so called deviation of a hypergraph. They proved that having small deviation is equivalent to a variety of other properties that describe quasi-randomness. In this paper, we consider the concept of discrepancy for k-uniform hypergraphs with an arbitrary constant density d (0 < d < 1) and prove that the condition of having asymptotically vanishing discrepancy is equivalent to several other quasi-random properties of H, similar to the ones introduced by Chung and Graham. In particular, we prove that the correct 'spectrum' of the s-vertex subhypergraphs is equivalent to quasi-randomness for any s ≥ 2k. Our work may be viewed as a continuation of the work of Chung and Graham, although our proof techniques are different in certain important parts.

Erdös-Hajnal conjecture states that for every undirected graph H there exists (H) > 0 such that every undirected graph on n vertices that does not contain H as an induced subgraph contains a clique or a stable set of size at least n (H). This conjecture has a directed equivalent version stating that for every tournament H there exists (H) > 0 such that every H−free n−vertex tournament T contains a transitive subtournament of order at least n (H). This conjecture is known to hold for a few infinite families of tournaments [1, 2, 6]. In this paper we construct two new infinite families of tournaments-the family of so-called galaxies with spiders and the family of so-called asterisms, and we prove the correctness of the conjecture for these two families.

Discrete & Computational Geometry, 2013

We study a high-dimensional analog for the notion of an acyclic (aka transitive) tournament. We give upper and lower bounds on the number of d-dimensional n-vertex acyclic tournaments. In addition, we prove that every n-vertex d-dimensional tournament contains an acyclic subtournament of Ω(log 1/d n) vertices and the bound is tight. This statement for tournaments (i.e., the case d = 1) is a well-known fact. We indicate a connection between acyclic high-dimensional tournaments and Ramsey numbers of hypergraphs. We investigate as well the interrelations among various other notions of acyclicity in high-dimensional tournaments. These include combinatorial, geometric and topological concepts.

We study the Maker-Breaker tournament game played on the edge set of a given graph $G$. Two players, Maker and Breaker claim unclaimed edges of $G$ in turns, and Maker wins if by the end of the game she claims all the edges of a pre-defined goal tournament. Given a tournament $T_k$ on $k$ vertices, we determine the threshold bias for the $(1:b)$ $T_k$-tournament game on $K_n$. We also look at the $(1:1)$ $T_k$-tournament game played on the edge set of a random graph ${\mathcal{G}_{n,p}}$ and determine the threshold probability for Maker&#39;s win. We compare these games with the clique game and discuss whether a random graph intuition is satisfied.

Random Structures &amp; Algorithms, 2011

We study quasi‐random properties of k‐uniform hypergraphs. Our central notion is uniform edge distribution with respect to large vertex sets. We will find several equivalent characterisations of this property and our work can be viewed as an extension of the well known Chung‐Graham‐Wilson theorem for quasi‐random graphs.Moreover, let Kk be the complete graph on k vertices and M(k) the line graph of the graph of the k‐dimensional hypercube. We will show that the pair of graphs (Kk,M(k)) has the property that if the number of copies of both Kk and M(k) in another graph G are as expected in the random graph of density d, then G is quasi‐random (in the sense of the Chung‐Graham‐Wilson theorem) with density close to d. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011

Electronic Notes in Discrete Mathematics, 2017

An n-vertex graph G of edge density p is considered to be quasirandom if it shares several important properties with the random graph Gpn, pq. A well-known theorem of Chung, Graham and Wilson states that many such 'typical' properties are asymptotically equivalent and, thus, a graph G possessing one such property automatically satisfies the others. In recent years, work in this area has focused on uncovering more quasirandom graph properties and on extending the known results to other discrete structures. In the context of hypergraphs, however, one may consider several different notions of quasirandomness. A complete description of these notions has been provided recently by Towsner, who proved several central equivalences using an analytic framework. We give short and purely combinatorial proofs of the main equivalences in Towsner's result.

Journal of The American Mathematical Society - J AMER MATH SOC, 1991