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.
Bibliographical noteFunding Information:
This research was supported by the Israel Science Foundation, Israel (Grant No. 1082/16).
© 2018 Elsevier Ltd
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics