On the minimum number of arcs in $4$-dicritical oriented graphs

Abstract: The dichromatic number

v e c c h i (D)

of a digraph

D

is the minimum number of colours needed to colour the vertices of a digraph such that each colour class induces an acyclic subdigraph. A digraph

D

is

k

-dicritical if

v e c c h i (D) = k

and each proper subdigraph

H

of

D

satisfies

v e c c h i (H) < k

. For integers

k

and

n

, we define

d_{k} (n)

(respectively

o_{k} (n)

) as the minimum number of arcs possible in a

k

-dicritical digraph (respectively oriented graph). Kostochka and Stiebitz have shown that

d_{4} (n) g e q f r a c 103 n - f r a c 43

. They also conjectured that there is a constant

c

such that

o_{k} (n) g e q c d_{k} (n)

for

k g e q 3

and

n

large enough. This conjecture is known to be true for

k = 3

(Aboulker et al.). In this work, we prove that every

4

-dicritical oriented graph on

n

vertices has at least

(f r a c 103 + f r a c 151) n - 1

arcs, showing the conjecture for

k = 4

. We also characterise exactly the

k

-dicritical digraphs on

n

vertices with exactly

f r a c 103 n - f r a c 43

arcs.