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

Abstract

It is known that every graph of sufficiently large chromatic number and bounded clique number contains, as an induced subgraph, a subdivision of any fixed forest, and a subdivision of any fixed cycle. Equivalently, forests and triangles are pervasive, where H is pervasive (in some class of graphs) if for all s>0, every graph in the class with bounded clique number and sufficiently large chromatic number contains an induced subdivision of H, with every edge subdivided at least s times. Which other graphs are pervasive? Chalopin, Esperet, Li and Ossona de Mendez proved that every such graph is a forest of lanterns: roughly, the blocks are lanterns (graphs obtained from a tree by adding one extra vertex), and there are rules about how blocks fit together. It is not known whether every forest of lanterns is pervasive; but in another paper two of us prove that banana trees (multigraphs obtained from a forest by adding parallel edges) are pervasive, thus generalizing the two results above. This paper contains the first half of the proof, which works for any forest of lanterns, not just for banana trees. A class of graphs is r-controlled if for every graph in the class, its chromatic number is at most some function (determined by the class) of the largest chromatic number of an r-ball in the graph. In this paper we prove that for all r>1, every forest of lanterns is pervasive in every r-controlled class These results turn out particularly nicely when applied to string graphs (intersection graphs of sets of curves in the plane). A chandelier is a graph obtained from a tree by adding a vertex adjacent to its leaves. We prove that the class of string graphs is 2-controlled, and thus forests of lanterns are pervasive in this class. Furthermore, string graphs of sufficiently large chromatic number and bounded clique number contain any fixed chandelier as an induced subgraph.