Hamed Hatami

Δ + 300 is a bound on the adjacent vertex distinguishing edge chromatic number

An adjacent vertex distinguishing edge-coloring or an avd-coloring of a simple graph G is a proper edge-coloring of G such that no pair of adjacent vertices meets the same set of colors. We prove that every graph with maximum degree Δ and with no isolated edges has an avd-coloring with at most Δ + 300 colors, provided that Δ > 1020.

On the number of pentagons in triangle-free graphs

Using the formalism of flag algebras, we prove that every triangle-free graph G with n vertices contains at most (n/5)^5 cycles of length five. Moreover, the equality is attained only when n is divisible by five and G is the balanced blow-up of the pentagon. We also compute the maximal number of pentagons and characterize extremal graphs in the non-divisible case provided n is sufficiently large. This settles a conjecture made by Erdos in 1984.

Every locally characterized affine-invariant property is testable

Set F = Fp for any fixed prime p ≥ 2. An affine-invariant property is a property of functions over Fn that is closed under taking affine transformations of the domain. We prove that all affine-invariant properties having local characterizations are testable. In fact, we show a proximity-oblivious test for any such property cP, meaning that given an input function f, we make a constant number of queries to f, always accept if f satisfies cP, and otherwise reject with probability larger than a positive number that depends only on the distance between f and cP. More generally, we show that any affine-invariant property that is closed under taking restrictions to subspaces and has bounded complexity is testable. We also prove that any property that can be described as the property of decomposing into a known structure of low-degree polynomials is locally characterized and is, hence, testable. For example, whether a function is a product of two degree-

The inducibility of blow-up graphs

d

The scaling window for a random graph with a given degree sequence

polynomials, whether a function splits into a product of d linear polynomials, and whether a function has low rank are all examples of degree-structural properties and are therefore locally characterized. Our results depend on a new Gowers inverse theorem by Tao and Ziegler for low characteristic fields that decomposes any polynomial with large Gowers norm into a function of a small number of low-degree non-classical polynomials. We establish a new equidistribution result for high rank non-classical polynomials that drives the proofs of both the testability results and the local characterization of degree-structural properties.

Random cubic graphs are not homomorphic to the cycle of size 7

The blow-up of a graph is obtained by replacing every vertex with a finite collection of copies so that the copies of two vertices are adjacent if and only if the originals are. If every vertex is replaced with the same number of copies, then the resulting graph is called a balanced blow-up.We show that any graph which contains the maximum number of induced copies of a sufficiently large balanced blow-up of H is itself essentially a blow-up of H. This gives an asymptotic answer to a question in 2.

The Fractional Chromatic Number of Graphs of Maximum Degree at Most Three

We consider a random graph on a given degree sequence <i>D</i>, satisfying certain conditions. We focus on two parameters <i>Q</i> = <i>Q</i>(<i>D</i>),<i>R</i> = <i>R</i>(<i>D</i>). Molloy and Reed proved that <i>Q</i> = 0 is the threshold for the random graph to have a giant component. We prove that if |<i>Q</i>| = <i>O</i>(<i>n</i>^-1/3<i>R</i>^2/3) then, with high probability, the size of the largest component of the random graph will be of order Θ(<i>n</i>^2/3<i>R</i>^-1/3). If <i>Q</i> is asymptotically larger/smaller that <i>n</i>^-1/3<i>R</i>^2/3 then the size of the largest component is asymptotically larger/smaller than <i>n</i>^2/3<i>R</i>^-1/3. In other words, we establish that |<i>Q</i>| = <i>O</i>(<i>n</i>^-1/3<i>R</i>^2/3) is the scaling window.

On the size of the minimum critical set of a Latin square

We prove that a random cubic graph almost surely is not homomorphic to a cycle of size 7. This implies that there exist cubic graphs of arbitrarily high girth with no homomorphisms to the cycle of size 7.

Decision trees and influences of variables over product probability spaces

This paper studies the fractional chromatic number of graphs with maximum degree at most 3. It is proved that if

Graph properties, graph limits, and entropy

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