A short construction of highly chromatic digraphs without short cycles

A natural digraph analogue of the graph-theoretic concept of an ‘independent set’ is that of an ‘acyclic set’, namely a set of vertices not spanning a directed cycle. Hence a digraph analogue of a graph coloring is a decomposition of the vertex set into acyclic sets. In the spirit of a famous theorem of P. Erdős [Graph theory and probability, Canad. J. Math. 11 (1959), 34–38], it was shown probabilistically in [D. Bokal et al., The circular chromatic number of a digraph, J. Graph Theory 46 (2004), no. 3, 227–240] that there exist digraphs with arbitrarily large girth and chromatic number. Here we give a construction of such digraphs. In [2], it is shown that the coloring theory for digraphs is similar to the coloring theory for graphs when stable sets are replaced by acyclic sets and homomorphisms are replaced by ‘acyclic homomorphisms’. One of the results therein asserts the existence of digraphs with arbitrarily large girth and (digraph) chromatic number. This, of course, is analogous to the seminal theorem of Erdős [3] on graphs with arbitrarily large girth and chromatic number, and it is likewise proved probabilistically, whence non-constructively. It is worth noting that although many results about digraph coloring theory are generalizations of results about graphs, the aforementioned result in [2] is not a generalization of Erdős’ theorem because the relationship between independent sets and cycles in graphs is different from the relationship between acyclic sets and directed cycles in digraphs. In this note, we construct digraphs with arbitrarily large girth and chromatic number. In fact, the construction strengthens the result in [2] because it produces a digraph with girth k and chromatic number n for each pair k, n of integers exceeding one. It is also of interest that unlike the analogous graph constructions in [4, 5, 6], our construction is primitively recursive in n. Received by the editors February 14, 2014, and in revised form March 11, 2014. 2010 Mathematics Subject Classification. 05C20, 05C15, 68R10.