Acyclic subgraphs with high chromatic number

Safwat Nassar, Raphael Yuster

Research output: Contribution to journalArticlepeer-review

Abstract

For an oriented graph G, let f(G) denote the maximum chromatic number of an acyclic subgraph of G. Let f(n) be the smallest integer such that every oriented graph G with chromatic number larger than f(n) has f(G)>n. Let g(n) be the smallest integer such that every tournament G with more than g(n) vertices has f(G)>n. It is straightforward that Ω(n)≤g(n)≤f(n)≤n2. This paper provides the first nontrivial lower and upper bounds for g(n). In particular, it is proved that [Formula presented]n8∕7≤g(n)≤n2−(2−[Formula presented])n+2. It is also shown that f(2)=3, i.e. every orientation of a 4-chromatic graph has a 3-chromatic acyclic subgraph. Finally, it is shown that a random tournament G with n vertices has f(G)=Θ([Formula presented]) whp.

Original languageEnglish
Pages (from-to)11-18
Number of pages8
JournalEuropean Journal of Combinatorics
Volume75
DOIs
StatePublished - Jan 2019

Bibliographical note

Funding Information:
This research was supported by the Israel Science Foundation, Israel (Grant No. 1082/16).

Publisher Copyright:
© 2018 Elsevier Ltd

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics

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