Minimum number of arcs in $k$-critical digraphs with order at most $2k-1$
Lucas Picasarri-Arrieta, Michael Stiebitz
TL;DR
The paper addresses the extremal problem for digraph colorings by studying the minimum number of arcs in $k$-critical digraphs of order $n$, where $\vec{\chi}(D)=k$. It extends Gallai’s classical graph-theoretic results to digraphs using the framework of the dichromatic number, the Dirac join decomposition, and structural bounds for near-critical order. The main contribution is a tight formula: for $n=k+p$ with $2\le p\le k-1$, the minimum number of arcs is $\overrightarrow{\mathrm{ext}}(k,n)=2\Big(\binom{n}{2}-(p^2+1)\Big)$, along with an exact description of the extremal digraphs achieving equality. The work also unifies decomposable and indecomposable critical digraphs, leveraging results by Aboulker–Vermande and Stehlík, and thereby advances the understanding of digraph colorings and their extremal structure in parallel with Gallai’s classical theory for graphs.
Abstract
The dichromatic number $\vecχ(D)$ of a digraph $D$ is the least integer $k$ for which $D$ has a coloring with $k$ colors such that there is no monochromatic directed cycle in $D$. The digraphs considered here are finite and may have antiparallel arcs, but no parallel arcs. A digraph $D$ is called $k$-critical if each proper subdigraph $D'$ of $D$ satisfies $\vecχ(D')<\vecχ(D)=k$. For integers $k$ and $n$, let $\overrightarrow{\mathrm{ext}}(k,n)$ denote the minimum number of arcs possible in a $k$-critical digraph of order $n$. It is easy to show that $\overrightarrow{\mathrm{ext}}(2,n)=n$ for all $n\geq 2$, and $\overrightarrow{\mathrm{ext}}(3,n)\geq 2n$ for all possible $n$, where equality holds if and only if $n$ is odd and $n\geq 3$. As a main result we prove that if $n, k$ and $p$ are integers with $n=k+p$ and $2\leq p \leq k-1$, then $\overrightarrow{\mathrm{ext}}(k,n)=2({\binom{n}{2}} - (p^2+1))$, and we give an exact characterisation of $k$-critical digraphs for which equality holds. This generalizes a result about critical graphs obtained in 1963 by Tibor Gallai.