# Various bounds on the minimum number of arcs in a k-dicritical digraph

The dichromatic number χ⃗(G) of a digraph G is the least integer k such that G can be partitioned into k acyclic digraphs. A digraph is k-dicritical if χ⃗(G) = k and each proper subgraph H of G satisfies χ⃗(H) ≤ k-1. cycle of length 2. We prove various bounds on the minimum number of arcs in a k-dicritical digraph, a structural result on k-dicritical digraphs and a result on list-dicolouring. We characterise 3-dicritical digraphs G with (k-1)|V(G)| + 1 arcs. For k ≥ 4, we characterise k-dicritical digraphs G on at least k+1 vertices and with (k-1)|V(G)| + k-3 arcs, generalising a result of Dirac. We prove that, for k ≥ 5, every k-dicritical digraph G has at least (k-1/2 - 1/(k-1)) |V(G)| - k(1/2 - 1/(k-1)) arcs, which is the best known lower bound. We prove that the number of connected components induced by the vertices of degree 2(k-1) of a k-dicritical digraph is at most the number of connected components in the rest of the digraph, generalising a result of Stiebitz. Finally, we generalise a Theorem of Thomassen on list-chromatic number of undirected graphs to list-dichromatic number of digraphs.

• 16 publications
• 6 publications
research
07/03/2022

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

The dichromatic number (D) of a digraph D is the least integer k such th...
research
11/15/2022

### Acyclic colourings of graphs with obstructions

Given a graph G, a colouring of G is acyclic if it is a proper colouring...
research
06/25/2018

### Even Longer Cycles in Essentially 4-Connected Planar Graphs

A planar graph is essentially 4-connected if it is 3-connected and every...
research
11/06/2019

### On the density of critical graphs with no large cliques

A graph G is k-critical if χ(G) = k and every proper subgraph of G is (k...
research
06/18/2020

### New Bounds for the Vertices of the Integer Hull

The vertices of the integer hull are the integral equivalent to the well...
research
12/04/2018

### Colouring triangle-free graphs with local list sizes

We prove two distinct and natural refinements of a recent breakthrough r...