The Gyárfás-Sumner conjecture asserts that if (Formula presented.) is a tree then every graph with bounded clique number and very large chromatic number contains (Formula presented.) as an induced subgraph. This is still open, although it has been proved for a few simple families of trees, including trees of radius two, some special trees of radius three, and subdivided stars. These trees all have the property that their vertices of degree more than two are clustered quite closely together. In this paper, we prove the conjecture for two families of trees which do not have this restriction. As special cases, these families contain all double-ended brooms and two-legged caterpillars.
All Science Journal Classification (ASJC) codes
- Geometry and Topology