On the minimum number of edges giving maximum oriented chromatic number

Abstract

We show that the minimum number of edges in a graph on n vertices with oriented chromatic number n is (1 + o(1))n log2 n. In 1995, in a conversation with the French member of the set of the authors of this note, Pal Erdős asked about the minimal number of edges a graph on n vertices with oriented chromatic number n can have. During the conference on the Future of Discrete Mathematics in the cosy but fruitful atmosphere of the Stǐŕin Castle we found an elementary answer to this question which we present below. 1 Novosibirsk State University, Novosibirsk, Russia 630090. Research partially supported by the grant 96-01-01614 of the Russian Foundation for Fundamental Research and by the Cooperative Grant Award RM1-181 of the US Civilian Research and Development Foundation. Department of Discrete Mathematics, Adam Mickiewicz University, 60-769 Poznan, Poland. Research partially supported by KBN grant 2 P03A 023 09. Mathematical Institute of the Hungarian Academy of Sciences, P.O.B.127, Budapest H-1364, Hungary. Research partially supported by the Hungarian National Foundation for Scientific Research (OTKA) Grant Nos. F023442 and T016386. LaBRI, Universite Bordeaux I, 33405 Talence Cedex, France. Research partially supported by the Barrande Grant no. 97137. After the final version of this note had been sent to the publisher we were informed that a very similar result had been proved independently by Z. Furedi, P. Horak, C. M. Pareek and X. Zhu in the paper Minimal oriented graphs of diameter 2, which is to appear in Graphs and Combinatorics.