# Past Combinatorial Theory Seminar

**Gyárfás** conjectured in 1985 that if $G$ is a graph with no induced cycle of odd length at least 5, then the chromatic number of $G$ is bounded by a function of its clique number. We prove this conjecture. Joint work with Paul Seymour.

More than twenty years ago Erdős conjectured that a triangle-free graph $G$ of chromatic number $k$ contains cycles of at least $k^{2−o(1)}$ different lengths. In this talk we prove this conjecture in a stronger form, showing that every such $G$ contains cycles of $ck^2\log k$ consecutive lengths, which is tight. Our approach can be also used to give new bounds on the number of different cycle lengths for other monotone classes of $k$-chromatic graphs, i.e., clique-free graphs and graphs without odd cycles.

Joint work with A. Kostochka and J. Verstraete.

We study the properties of finite graphs in which the ball of radius $r$ around each vertex induces a graph isomorphic to some fixed graph $F$. (Such a graph is said to be $r$-locally-$F$.) This is a natural extension of the study of regular graphs, and of the study of graphs of constant link. We focus on the case where $F$ is $\mathbb{L}^d$, the $d$-dimensional integer lattice. We obtain a characterisation of all the finite graphs in which the ball of radius $3$ around each vertex is isomorphic to the ball of radius $3$ in $\mathbb{L}^d$, for each integer $d$. These graphs have a very rigidly proscribed global structure, much more so than that of $(2d)$-regular graphs. (They can be viewed as quotient lattices in certain 'flat orbifolds'.) Our results are best possible in the sense that '3' cannot be replaced with '2'. Our proofs use a mixture of techniques and results from combinatorics, algebraic topology and group theory. We will also discuss some results and open problems on the properties of a random n-vertex graph which is $r$-locally-$F$. This is all joint work with Itai Benjamini (Weizmann Institute of Science).

In the Erdös-Rényi random graph process, starting from an empty graph, in each
step a new random edge is added to the evolving graph. One of its most
interesting features is the `percolation phase transition': as the ratio of the
number of edges to vertices increases past a certain critical density, the
global structure changes radically, from only small components to a single
giant component plus small ones.

In this talk we consider Achlioptas processes, which have become a key example
for random graph processes with dependencies between the edges. Starting from
an empty graph these proceed as follows: in each step two potential edges are
chosen uniformly at random, and using some rule one of them is selected and
added to the evolving graph. We discuss why, for a large class of rules, the
percolation phase transition is qualitatively comparable to the classical
Erdös-Rényi process.

Based on joint work with Oliver Riordan.