A hole in a graph is an induced cycle of length at least four, and an odd hole is a hole of odd length. A famous conjecture of A. Gyárfás [1] from 1985 asserts:
Conjecture 1: For all integers $k,l$ there exists $n(k,l)$ such that every graph $G$ with no clique of carnality more than $k$ and no odd hole of length more than $l$ has chromatic number at most $n(k,l)$.

In other words, the conjecture states that the family of graphs with no long odd holes is $\chi$-bounded. Little progress was made on this problem until recently Scott and Seymour proved that Conjecture 1 is true for all pairs $(k,l)$ when $l=3$ (thus excluding all odd holes guarantees $\chi$-boundedness) [3].
No other cases have been settled, and here we focus on the case $k=2$. We resolve the first open case, when $k=2$ and $l=5$, proving that
Theorem 1. Every graph with no triangle and no odd hole of length $>5$ is $82200$-colorable.

Conjecture 1 has a number of other interesting special cases that still remain open; for instance
* Conjecture: For all $l$ every triangle-free graph $G$ with sufficiently large chromatic number has an odd hole of length more than $l$;
* Conjecture: For all $k,l$ every graph with no clique of size more than $k$ and sufficiently large chromatic number has a hole of length more than $l$.

We prove both these statements with the additional assumption that $G$ contains no $5$-hole. (The latter one was proved, but not published, by Scott earlier, improving on [2]). All the proofs follows a similar outline. We start with a leveling of a graph with high chromatic number, that is a classification of the vertices by their distance from a fixed root. Then the graph undergoes several rounds of "trimming" that allows us to focus on a subgraph $M$ with high chromatic number that is, in some sense, minimal. We also ensure that certain pairs of vertices with a neighbor in $M$ can be joined by a path whose interior is anticomplete to $M$. It is now enough to find two long paths between some such pair of vertices, both with interior in $M$ and of lengths of different parity, to obtain a long odd hole.[-]

Maria Chudnovsky is a professor in the department of mathematics at Princeton University. She grew up in Russia and Israel, studying at the Technion and received her Ph.D. in 2003 from Princeton under the supervision of Paul Seymour. She moved to Columbia after being a Clay Mathematics Institute research fellow and assistant professor at Princeton. Chudnovsky's contributions to graph theory include the proof of the strong perfect graph theorem with Robertson, Seymour and Thomas characterizing perfect graphs as being exactly the graphs with no odd induced cycles of length at least 5 or their complements. Other research contributions of Chudnovsky include co-authorship of the first polynomial time algorithm for recognizing perfect graphs and of a structural characterization of the claw-free graphs.[-]

The problem of testing if a graph can be colored with a given number $k$ of colors is $NP$-complete for every $k>2$. But what if we have more information about the input graph, namely that some fixed graph $H$ is not present in it as an induced subgraph? It is known that the problem remains $NP$-complete even for $k=3$, unless $H$ is the disjoint union of paths. We consider the following two questions: 1) For which graphs $H$ is there a polynomial time algorithm to 3-color (or in general $k$-color) an $H$-free graph? 2) For which graphs $H$ are there finitely many 4-critical $H$-free graphs? This talk will survey recent progress on these questions, and in particular give a complete answer to the second one.[-]