Induced subgraphs of graphs with large chromatic number. VI. Banana trees

Abstract: We investigate which graphs H have the property that in every graph with bounded clique number and sufficiently large chromatic number, some induced subgraph is isomorphic to a subdivision of H. In an earlier paper, one of us proved that every tree has this property; and in another earlier paper with M. Chudnovsky, we proved that every cycle has this property. Here we give a common generalization. Say a banana is the union of a set of paths all with the same ends but otherwise disjoint. We prove that if H is obtained from a tree by replacing each edge by a banana then H has the property mentioned. We also find some other multigraphs with the same property.