The Minimum Number of Edges in 4-Critical Digraphs of Given Order

Abstract

The dichromatic number $\chi (D)$ of a digraph D, introduced by Neumann-Lara in the 1980s, is the least integer k for which D has a coloring with k colors such that each vertex receives a color and no directed cycle of D is monochromatic. The digraphs considered here are finite and may have antiparalell arcs, but no parallel arcs. A digraph D is k-critical if each proper subdigraph $D'$ of D satisfies $\chi (D')<\chi (D)=k$. For integers k and n, let $d_k(n)$ denote the minimum number of arcs possible in a k-critical digraph of order n. It is easy to show that $d_2(n)=n$ for all $n\ge 2$, and $d_3(n)\ge 2n$ for all possible n, where equality holds if and only if n is odd and $n\ge 3$. As a main result we prove that $d_4(n)\ge \lceil (10n-4)/3\rceil$ for all $n\ge 4$ and $n\not =5$ where equality holds if $n \equiv 1 \, \mathrm {(mod} \; 3 \mathrm {)}$ or $n \equiv 2 \, \mathrm {(mod} \; 3 \mathrm {)}$. As a consequence, we obtain that for each surface $\mathbf{S}$ the number of 4-critical oriented graphs that can be embedded on $\mathbf{S}$ is finite.