On the chromatic number of random Cayley graphs

Let G be an abelian group of cardinality N, where (N,6) = 1, and let A be a random subset of G. Form a graph Gamma_A on vertex set G by joining x to y if and only if x + y is in A. Then, almost surely as N tends to infinity, the chromatic number chi(Gamma_A) is at most (1 + o(1))N/2 log_2 N. This is asymptotically sharp when G = Z/NZ, N prime. Presented at the conference in honour of Bela Bollobas on his 70th birthday, Cambridge August 2013.Comment: 26 pages, revised following referee's repor

Distinguishing Chromatic Number of Random Cayley graphs

The \textit{Distinguishing Chromatic Number} of a graph $G$, denoted $\chi_D(G)$, was first defined in \cite{collins} as the minimum number of colors needed to properly color $G$ such that no non-trivial automorphism $\phi$ of the graph $G$ fixes each color class of $G$. In this paper, we consider random Cayley graphs $\Gamma(A,S)$ defined over certain abelian groups $A$ and show that with probability at least $1-n^{-\Omega(\log n)}$ we have, $\chi_D(\Gamma)\le\chi(\Gamma) + 1$.Comment: 11 page

Perfect Matchings as IID Factors on Non-Amenable Groups

We prove that in every bipartite Cayley graph of every non-amenable group, there is a perfect matching that is obtained as a factor of independent uniform random variables. We also discuss expansion properties of factors and improve the Hoffman spectral bound on independence number of finite graphs.Comment: 16 pages; corrected missing reference in v

On the quantum chromatic number of a graph

We investigate the notion of quantum chromatic number of a graph, which is the minimal number of colours necessary in a protocol in which two separated provers can convince an interrogator with certainty that they have a colouring of the graph. After discussing this notion from first principles, we go on to establish relations with the clique number and orthogonal representations of the graph. We also prove several general facts about this graph parameter and find large separations between the clique number and the quantum chromatic number by looking at random graphs. Finally, we show that there can be no separation between classical and quantum chromatic number if the latter is 2, nor if it is 3 in a restricted quantum model; on the other hand, we exhibit a graph on 18 vertices and 44 edges with chromatic number 5 and quantum chromatic number 4.Comment: 7 pages, 1 eps figure; revtex4. v2 has some new references; v3 furthe small improvement

Pseudo-random graphs

Random graphs have proven to be one of the most important and fruitful concepts in modern Combinatorics and Theoretical Computer Science. Besides being a fascinating study subject for their own sake, they serve as essential instruments in proving an enormous number of combinatorial statements, making their role quite hard to overestimate. Their tremendous success serves as a natural motivation for the following very general and deep informal questions: what are the essential properties of random graphs? How can one tell when a given graph behaves like a random graph? How to create deterministically graphs that look random-like? This leads us to a concept of pseudo-random graphs and the aim of this survey is to provide a systematic treatment of this concept.Comment: 50 page