Subgraphs of large connectivity and chromatic number

Induced subgraphs of graphs with large chromatic number. XI. Orientations

Fix an oriented graph H, and let G be a graph with bounded clique number and very large chromatic number. If we somehow orient its edges, must there be an induced subdigraph isomorphic to H? Kierstead and Rödl [12] raised this question for two specific kinds of digraph H: the three-edge path, with the first and last edges both directed towards the interior; and stars (with many edges directed o...

Triangle-free subgraphs with large fractional chromatic number

It is well known that for any k and g, there is a graph with chromatic number at least k and girth at least g. In 1970’s, Erdős and Hajnal conjectured that for any numbers k and g, there exists a number f(k, g), such that every graph with chromatic number at least f(k, g) contains a subgraph with chromatic number at least k and girth at least g. In 1978, Rödl proved the case for g = 4 and arbit...

Size, chromatic number, and connectivity

Induced subgraphs of graphs with large chromatic number. V. Chandeliers and strings

A “string graph” is the intersection graph of a set of curves in the plane. It is known [9] that there are string graphs with clique number two and chromatic number arbitrarily large, and in this paper we study the induced subgraphs of such graphs. Let us say a graph H is “pervasive” (in some class of graphs) if for all l ≥ 1, and in every graph in the class of bounded clique number and suffici...

Induced subgraphs of graphs with large chromatic number. VII. Gyárfás’ complementation conjecture

A class of graphs is χ-bounded if there is a function f such that χ(G) ≤ f(ω(G)) for every induced subgraph G of every graph in the class, where χ, ω denote the chromatic number and clique number of G respectively. In 1987, Gyárfás conjectured that for every c, if C is a class of graphs such that χ(G) ≤ ω(G) + c for every induced subgraph G of every graph in the class, then the class of complem...