In 1973, P. Erdös conjectured that for each kε2, there exists a constant c k so that if G is a graph on n vertices and G has no odd cycle with length less than c k n 1/k, then the chromatic number of G is at most k+1. Constructions due to Lovász and Schriver show that c k, if it exists, must be at least 1. In this paper we settle Erdös' conjecture in the affirmative. We actually prove a stronger result which provides an upper bound on the chromatic number of a graph in which we have a bound on the chromatic number of subgraphs with small diameter.
- AMS subject classification (1980): 05C15
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
- Computational Mathematics