Chromatic number and subtrees of graphs

Abstract

Let G and H be two graphs. We say that G induces H if G has an induced subgraph isomorphic to H: A. Gyárfás and D. Sumner, independently, conjectured that, for every tree T. there exists a function f _T ; called binding function, depending only on T with the property that every graph G with chromatic number f _T (ω(G)) induces T. A. Gyárfás, E. Szemerédi and Z. Tuza confirmed the conjecture for all trees of radius two on triangle-free graphs, and H. Kierstead and S. Penrice generalized the approach and the conclusion of A. Gyárfás et al. onto general graphs. A. Scott proved an interesting topological version of this conjecture asserting that for every integer k and every tree T of radius r, every graph G with ω(G) ⩽ k and sufficient large chromatic number induces a subdivision of T of which each edge is subdivided at most O(14^r-1(r - 1)!) times. We extend the approach of A. Gyárfás and present a binding function for trees obtained by identifying one end of a path and the center of a star. We also improve A. Scott's upper bound by modifying his subtree structure and partition technique, and show that for every integer k and every tree T of radius r, every graph with ω(G) ⩽ k and sufficient large chromatic number induces a subdivision of T of which each edge is subdivided at most O(6^r−2) times.