On the minimum number of arcs in $k$-dicritical oriented graphs

Abstract

The dichromatic number χ(D) of a digraph D is the least integer k such that D can be partitionedAn oriented graph is a digraph with no directed cycle of length 2. For integers k and n, we denote by o k (n) the minimum number of edges of a k-critical oriented graph on n vertices (with the convention o k (n) = +∞ if there is no k-dicritical oriented graph of order n). The main result of this paper is a proof that o 3 (n) ≥ 7n+2 3 together with a construction witnessing that o 3 (n) ≤ 5n 2 for all n ≥ 12. We also give a construction showing that for all sufficiently large n and all k ≥ 3, o k (n) < (2k − 3)n, disproving a conjecture of Hoshino and Kawarabayashi. Finally, we prove that, for all k ≥ 2, o k (n) ≥ k − 3 4 − 1 4k−6 n + 3 4(2k−3) , improving the previous best known lower bound of Bang-Jensen, Bellitto, Schweser and Stiebitz.

On the Minimum Number of Arcs in 4-Dicritical Oriented Graphs

“…Conjecture 7 has been confirmed for k = 3 by Aboulker et al [2] and for k = 4 by the first and third authors, and Rambaud [16].…”

“…Already the minimum number n k of vertices of a k-dicritical oriented graph is unknown except for small values of k : clearly n 2 = 3 ; Neumann Lara [24] proved n 3 = 7 and n 4 = 11 ; Bellitto et al [5] recently established n 5 = 19. As observed by Aboulker et al [2] using a lemma of Hoshino and Kawarabayashi [17], there exists a smallest integer p k such that there exists a k-dicritical oriented graph on n vertices for any n ≥ p k . Moreover, while p 3 = n 3 = 7, they showed that p 4 = n 4 because there is no 4-dicritical oriented graph on 12 vertices.…”

“…It is also interesting to determine the maximum number of edges in a k-critical graph. Erdős [12] asked, for every fixed k ≥ 4, whether there exists a constant c k > 0 such that there exist arbitrarily large k-critical graphs G with at least c k • n(G) 2 edges. This was proved by Dirac [8] when k ≥ 6 and then by Toft [27] when k ∈ {4, 5}.…”

“…Let D be a digraph and uv be an arc of D. A uv-colouring of D is a 2-colouring φ : V (D) − → [2] such that:…”

“…Proof. As D is 3-dicritical, there is a 2-dicolouring φ : V (D) → [2] of D \ uv. By symmetry, we may suppose φ(u) = 1.…”