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

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, every forest is pervasive, and K₃ is pervasive, in the class of all graphs, where we say a graph H is “pervasive” (in some class of graphs) if for all ℓ≥1, every graph in the class of bounded clique number and sufficiently large chromatic number has an induced subgraph that is a subdivision of H, in which every edge of H is replaced by a path of at least ℓ edges. Which other graphs are pervasive? It was proved by Chalopin, Esperet, Li and Ossona de Mendez that every such graph is a “forest of lanterns”: roughly, every block is a “lantern”, a graph obtained from a tree by adding one extra vertex, and there are rules about how the blocks fit together. It is not known whether every forest of lanterns is pervasive in the class of all graphs; but in another paper two of us prove that all “banana trees” are pervasive, that is, multigraphs obtained from a forest by adding parallel edges, 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. Say a class of graphs is “ρ-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 a ρ-ball in the graph. In this paper we prove that for every ρ≥2, and for every ρ-controlled class, every forest of lanterns is pervasive in this class. These results turn out particularly nicely when applied to string graphs. A “chandelier” is a special lantern, a graph obtained from a tree by adding a vertex adjacent to precisely the leaves of the tree. A “string graph” is the intersection graph of a set of curves in the plane. There are string graphs with clique number two and chromatic number arbitrarily large. We prove that the class of string graphs is 2-controlled, and consequently every forest of lanterns is pervasive in this class; but in fact something stronger is true, that every string graph of sufficiently large chromatic number and bounded clique number contains each fixed chandelier as an induced subgraph (not just as a subdivision); and the same for most forests of chandeliers (there is an extra condition on how the blocks are attached together).