Radius Three Trees in Graphs with Large Chromatic Number

Abstract

A class Γ of graphs is χ-bounded if there exists a function f such that χ (G) ≤ f (ω (G)) for all graphs G ∈ Γ, where χ denotes chromatic number and ω denotes clique number. Gyárfás and Sumner independently conjectured that, for any tree T , the class Forb (T ), consisting of graphs that do not contain T as an induced subgraph, is χ-bounded. The first author and Penrice showed that this conjecture is true for any radius two tree. Here we use the work of several authors to show that the conjecture is true for radius three trees obtained from radius two trees by making exactly one subdivision in every edge adjacent to the root. These are the only trees with radius greater than two, other than subdivided stars, for which the conjecture is known to be true.

Excluding cycles with a fixed number of chords

“…It is actually conjectured in [6] that Forb (H) is χ-bounded if and only if H is a forest. The deeper results concerning this conjecture are certainly results of Kierstead and Penrice [8] and Kierstead and Zhu [9] proving that the conjecture holds for every tree of radius at most 2 and several trees of radius 3. To get out from this conjecture, we need to forbid a class of graph H such that H contains graphs with arbitrarily large girth.…”

Unavoidable Subtournaments in Large Tournaments with No Homogeneous Sets

“…Proof of Theorem 1. 13. We prove that for a tournament T , if T is {D n , U 3 }free for some n ≥ 2, then T is 3 n−2 -colorable.…”

“…This conjecture is known to be true for several classes of forests [11,12,13,20,5], but is mostly wide open.…”

Substitution and χ -boundedness

“…We observe that if v i , v j ∈ V F are adjacent, then ω i + ω j ≤ ω; since F contains no isolated vertices, it follows that ω i ≤ ω − 1 for all i ∈ {1, ..., n}. 16 , +∞) and ω 2 ≥ 2. We now define:…”