Serguei Norine

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.

Rank of divisors on tropical curves

We investigate, using purely combinatorial methods, structural and algorithmic properties of linear equivalence classes of divisors on tropical curves. In particular, we confirm a conjecture of Baker asserting that the rank of a divisor D on a (non-metric) graph is equal to the rank of D on the corresponding metric graph, and construct an algorithm for computing the rank of a divisor on a tropical curve.

Exponentially many perfect matchings in cubic graphs

We show that every cubic bridgeless graph G has at least 2|V(G)|/3656 perfect matchings. This confirms an old conjecture of Lovasz and Plummer.

Undecidability of linear inequalities in graph homomorphism densities

The purpose of this article is to show that even the most elementary problems in asymptotic extremal graph theory can be highly non-trivial. We study linear inequalities between graph homomorphism densities. In the language of quantum graphs the validity of such an inequality is equivalent to the positivity of a corresponding quantum graph. Similar to the setting of polynomials, a quantum graph that can be represented as a sum of squares of labeled quantum graphs is necessarily positive. Lovasz asks whether the opposite is also true. We answer this question and also a related question of Razborov in the negative by introducing explicit valid inequalities that do not satisfy the required conditions. Our solution to these problems is based on a reduction from real multivariate polynomials and uses the fact that there are positive polynomials that cannot be expressed as sums of squares of polynomials. nIt is known that the problem of determining whether a multivariate polynomial is positive is decidable. Hence it is very natural to ask Is the problem of determining the validity of a linear inequality between homomorphism densities decidable? We give a negative answer to this question which shows that such inequalities are inherently difficult in their full generality. Furthermore we deduce from this fact that the analogue of Artins solution to Hilberts seventeenth problem does not hold in the setting of quantum graphs.

Non-three-colourable common graphs exist

A graph H is called common if the sum of the number of copies of H in a graph G and the number in the complement of G is asymptotically minimized by taking G to be a random graph. Extending a conjecture of Erdős, Burr and Rosta conjectured that every graph is common. Thomason disproved both conjectures by showing that K4 is not common. It is now known that in fact the common graphs are very rare. Answering a question of Sidorenko and of Jagger, Stovi-cek and Thomason from 1996 we show that the 5-wheel is common. This provides the first example of a common graph that is not three-colourable.

K6 minors in large 6-connected graphs

Jorgensen conjectured that every 6-connected graph with no K_6 minor has a vertex whose deletion makes the graph planar. We prove the conjecture for all sufficiently large graphs.

Voting in Agreeable Societies

When can a majority of voters find common ground, that is, a position they all agree upon? How does the shape of the political spectrum influence the outcome? When mathematical objects have a social interpretation, the associated theorems have social applications. In this article we give examples of situations where sets model preferences and develop extensions of classical theorems about convex sets, such as Hellys theorem, that can be used in the analysis of voting in agreeable societies.

The inducibility of blow-up graphs

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.

K6 minors in 6-connected graphs of bounded tree-width

Abstract We prove that every sufficiently large 6-connected graph of bounded tree-width either has a K 6 minor, or has a vertex whose deletion makes the graph planar. This is a step toward proving that the same conclusion holds for all sufficiently large 6-connected graphs. Jorgensen conjectured that it holds for all 6-connected graphs.

Small graph classes and bounded expansion

A class of simple undirected graphs is small if it contains at most n!@a^n labeled graphs with n vertices, for some constant @a. We prove that for any constants c,@e>0, the class of graphs with expansion bounded by the function f(r)=c^r^^^1^^^/^^^3^^^-^^^@e is small. Also, we show that the class of graphs with expansion bounded by 6@?3^r^l^o^g^(^r^+^e^) is not small.