Induced subgraphs of graphs with large chromatic number. III. Long holes

We prove a 1985 conjecture of Gy\'arf\'as that for all $k,\ell$, every graph with sufficiently large chromatic number contains either a complete subgraph with $k$ vertices or an induced cycle of length at least $\ell$

Induced subgraphs of graphs with large chromatic number. II. Three steps towards Gyarfas' conjectures

Gyarfas conjectured in 1985 that for all $k$, $l$, every graph with no clique of size more than $k$ and no odd hole of length more than $l$ has chromatic number bounded by a function of $k$ and $l$. We prove three weaker statements: (1) Every triangle-free graph with sufficiently large chromatic number has an odd hole of length different from five; (2) For all $l$, every triangle-free graph with sufficiently large chromatic number contains either a 5-hole or an odd hole of length more than $l$; (3) For all $k$, $l$, every graph with no clique of size more than $k$ and sufficiently large chromatic number contains either a 5-hole or a hole of length more than $l$

Induced subgraphs of graphs with large chromatic number. IV. Consecutive holes

A hole in a graph is an induced subgraph which is a cycle of length at least four. We prove that for every positive integer k, every triangle-free graph with sufficiently large chromatic number contains holes of k consecutive lengths

A survey of χ\chi-boundedness

If a graph has bounded clique number, and sufficiently large chromatic number, what can we say about its induced subgraphs? Andr\'as Gy\'arf\'as made a number of challenging conjectures about this in the early 1980's, which have remained open until recently; but in the last few years there has been substantial progress. This is a survey of where we are now

On hereditary graph classes defined by forbidding Truemper configurations: recognition and combinatorial optimization algorithms, and χ-boundedness results

Truemper configurations are four types of graphs that helped us understand the structure of several well-known hereditary graph classes. The most famous examples are perhaps the class of perfect graphs and the class of even-hole-free graphs: for both of them, some Truemper configurations are excluded (as induced subgraphs), and this fact appeared to be useful, and played some role in the proof of the known decomposition theorems for these classes. The main goal of this thesis is to contribute to the systematic exploration of hereditary graph classes defined by forbidding Truemper configurations. We study many of these classes, and we investigate their structure by applying the decomposition method. We then use our structural results to analyze the complexity of the maximum clique, maximum stable set and optimal coloring problems restricted to these classes. Finally, we provide polynomial-time recognition algorithms for all of these classes, and we obtain χ-boundedness results

A note on chromatic number and induced odd cycles

An odd hole is an induced odd cycle of length at least 5. Scott and Seymour confirmed a conjecture of Gyarfas and proved that if a graph G has no odd holes then chi(G) \u3c=( 2 omega(G)+2). Chudnovsky, Robertson, Seymour and Thomas showed that if G has neither K-4 nor odd holes then chi(G) \u3c= 4. In this note, we show that if a graph G has neither triangles nor quadrilaterals, and has no odd holes of length at least 7, then chi(G) \u3c= 4 and chi(G) \u3c= 3 if G has radius at most 3, and for each vertex u of G, the set of vertices of the same distance to u induces abipartite subgraph. This answers some questions in [17]