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

Frédéric Havet, Lucas Picasarri-Arrieta, Clément Rambaud

TL;DR

The paper addresses the minimum arc count in $4$-dicritical oriented graphs by developing a potential-method framework that accounts for digons via the parameter $T(D)$ and the potential $\rho(D)=(\frac{10}{3}+\varepsilon)n(D)-m(D)-\delta T(D)$. It introduces $4$-Ore digraphs as the extremal cases and uses Ore-compositions, collapsible subdigraphs, and discharging to obtain a global contradiction for non-extremal counterexamples, yielding a sharp bound $m(D)\ge\bigl(\frac{10}{3}+\frac{1}{51}\bigr)n(D)-1$ for $4$-dicritical oriented graphs; equality cases align with $4$-Ore digraphs. The authors also characterize when $m(D)=\frac{10}{3}n(D)-\frac{4}{3}$, showing these are exactly the $4$-Ore digraphs. Additionally, they prove an upper bound on $o_k(n)$ for fixed $k$, showing there are infinitely many $n$ with $o_k(n)\le(2k-\tfrac{7}{2})n$ via a constructive, recursive gadget framework, thereby substantiating the conjecture for $k=4$ and providing insights into the structure of $k$-dicritical oriented graphs.

Abstract

The dichromatic number $\vecχ(D)$ 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 $\vecχ(D) = k$ and each proper subdigraph $H$ of $D$ satisfies $\vecχ(H) < k$. 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 $d_4(n) \geq \frac{10}{3}n -\frac{4}{3}$. They also conjectured that there is a constant $c$ such that $o_k(n) \geq cd_k(n)$ for $k\geq 3$ 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 $(\frac{10}{3}+\frac{1}{51})n-1$ arcs, showing the conjecture for $k=4$. We also characterise exactly the $k$-dicritical digraphs on $n$ vertices with exactly $\frac{10}{3}n -\frac{4}{3}$ arcs.

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

TL;DR

The paper addresses the minimum arc count in -dicritical oriented graphs by developing a potential-method framework that accounts for digons via the parameter and the potential . It introduces -Ore digraphs as the extremal cases and uses Ore-compositions, collapsible subdigraphs, and discharging to obtain a global contradiction for non-extremal counterexamples, yielding a sharp bound for -dicritical oriented graphs; equality cases align with -Ore digraphs. The authors also characterize when , showing these are exactly the -Ore digraphs. Additionally, they prove an upper bound on for fixed , showing there are infinitely many with via a constructive, recursive gadget framework, thereby substantiating the conjecture for and providing insights into the structure of -dicritical oriented graphs.

Abstract

The dichromatic number of a digraph 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 is -dicritical if and each proper subdigraph of satisfies . For integers and , we define (respectively ) as the minimum number of arcs possible in a -dicritical digraph (respectively oriented graph). Kostochka and Stiebitz have shown that . They also conjectured that there is a constant such that for and large enough. This conjecture is known to be true for (Aboulker et al.). In this work, we prove that every -dicritical oriented graph on vertices has at least arcs, showing the conjecture for . We also characterise exactly the -dicritical digraphs on vertices with exactly arcs.
Paper Structure (4 sections, 22 theorems, 49 equations, 6 figures)

This paper contains 4 sections, 22 theorems, 49 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The 4-Ore digraphs and their properties
  3. Proof of Theorem \ref{['thm:main_thm_potential']}
  4. An upper bound on ok(n)

Key Result

Theorem 1

Every $k$-critical graph on $n$ vertices has at least $\frac{1}{2}(k-\frac{2}{k-1})n - \frac{k(k-3)}{2(k-1)}$ edges. For every $k$, this bound is tight for infinitely many values of $n$.

Figures (6)

  • Figure 1: An example of a 4-Ore digraph obtained by an Ore-composition of two smaller 4-Ore digraphs, with replaced digon $[x,y]$ and split vertex $z$.
  • Figure 2: A $4$-dicritical digraph $D$ together with an induced subdigraph $R$ of $D$ and $\varphi$ a $3$-dicolouring of $R$, the $\varphi$-identification $D_\varphi(R)$ of $R$ in $D$ and the dicritical extension $R'$ of $R$ with extender $W$ and core $X _W$. For clarity, the digons are represented by undirected edges.
  • Figure 3: The possible connected components of $D_6$.
  • Figure 4: An example of an out-chelou arc $xy$.
  • Figure 5: A 3-dicritical oriented graph with $\frac{5}{2}n$ arcs.
  • ...and 1 more figures

Theorems & Definitions (63)

  • Theorem 1: Kostochka and Yancey kostochka_ores_2014
  • Conjecture 2: Postle Postle-EJC17
  • Conjecture 3: Kostochka and Stiebitz kostochka_minimum_2020
  • Conjecture 4: Kostochka and Stiebitz kostochka_minimum_2020
  • Conjecture 5
  • Conjecture 6
  • Theorem 7
  • Theorem 7
  • Theorem 8
  • Corollary 9
  • ...and 53 more