The edge density of critical digraphs

Abstract

Let χ(G) denote the chromatic number of a graph G. We say that G is k-critical if χ(G)=k and χ(H) < k for every proper subgraph H ⊂ G. Over the years, many properties of k-critical graphs have been discovered, including improved upper and lower bounds for ||G||, the number of edges in a k-critical graph, as a function of |G|, the number of vertices. In this note, we analyze this edge density problem for directed graphs, where the chromatic number χ(D) of a digraph D is defined to be the fewest number of colours needed to colour the vertices of D so that each colour class induces an acyclic subgraph. For each k ≥ 3, we construct an infinite family of sparse k-critical digraphs for which $\left\| D \right\| < \left( {\tfrac{{k^2 - k + 1}} {2}} \right)\left| D \right|$ and an infinite family of dense k-critical digraphs for which $\left\| D \right\| > \left( {\tfrac{1} {2} - \tfrac{1} {{2^k - 1}}} \right)\left| D \right|^2$. One corollary of our results is an explicit construction of an infinite family of k-critical digraphs of digirth l, for any pair of integers k,l≥3. This extends a result by Bokal et al. who used a probabilistic approach to demonstrate the existence of one such digraph.