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 kcritical 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 ||D|| < ( k 2 −k+1 2 )|D| and an infinite family of dense k-critical digraphs for which ||D|| > 1 2 − 1 2 k−1 |D| 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.

Hajós and Ore Constructions for Digraphs

“…For the proof of the next theorem, recall that a k-coloring of a digraph D is a coloring of D, in which at most k colors are used. Statement (c) of the following theorem has already been mentioned in [13,Prop. 2 Theorem 2 (Hajós Construction) Let D = D 1 D 2 be the Hajós join of two disjoint non-empty digraphs D 1 and D 2 .…”

“…Let D be the digraph obtained from the union D 1 ∪ D 2 by adding all possible arcs in both directions between D 1 and D 2 , i.e., The Hajós join is a well-known tool for undirected graphs that can be used to create infinite families of k-critical graphs, see e. g. [8]. For digraphs, an equivalent construction was defined by Hoshino and Kawarabayashi in [13]. Let D 1 and D 2 be two disjoint digraphs and select an arc u 1 v 1 and an arc v 2 u 2 .…”

Extension of Gyarfas-Sumner conjecture to digraphs

“…Extending the notion of chromatic number, the acyclic chromatic number, or simply dichromatic number, of a digraph D, denoted χ(D), is defined to be the smallest number of colors required for an acyclic coloring of D. Being viewed as a natural generalization of the chromatic number to digraphs, the dichromatic number has been the center of attention recently. See [2,5,13,15,18,19] for examples of recent works on this subject.…”

Four proofs of the directed Brooks' Theorem

“…This simple fact permits to generalize results on the chromatic number of undirected graphs to digraphs via the dichromatic number. Such results have (recently) been found in various area of graph colouring such as extremal graph theory [5,20,21], algebraic graph theory [27], substructure forced by large dichromatic number [1,2,3,8,13,14,32], list dichromatic number [7,18], dicolouring digraphs on surfaces [4,25,30], flow theory [19,22], links between dichromatic number and girth [16,17,31].…”