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The 3-dicritical semi-complete digraphs

Frédéric Havet, Florian Hörsch, Lucas Picasarri-Arrieta

Abstract

A digraph is $3$-dicritical if it cannot be vertex-partitioned into two sets inducing acyclic digraphs, but each of its proper subdigraphs can. We give a human-readable proof that the number of 3-dicritical semi-complete digraphs is finite. Further, we give a computer-assisted proof of a full characterization of 3-dicritical semi-complete digraphs. There are eight such digraphs, two of which are tournaments. We finally give a general upper bound on the maximum number of arcs in a $3$-dicritical digraph.

The 3-dicritical semi-complete digraphs

Abstract

A digraph is -dicritical if it cannot be vertex-partitioned into two sets inducing acyclic digraphs, but each of its proper subdigraphs can. We give a human-readable proof that the number of 3-dicritical semi-complete digraphs is finite. Further, we give a computer-assisted proof of a full characterization of 3-dicritical semi-complete digraphs. There are eight such digraphs, two of which are tournaments. We finally give a general upper bound on the maximum number of arcs in a -dicritical digraph.
Paper Structure (16 sections, 28 theorems, 10 equations, 7 figures)

This paper contains 16 sections, 28 theorems, 10 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. Context
  4. Our results
  5. Useful lemmas
  6. A simple proof for finiteness
  7. The 3-dicritical semi-complete digraphs
  8. Maximum number of arcs in 3-dicritical digraphs
  9. Conclusion
  10. The tournaments T1,...,T4
  11. Code used in the proof of Theorem \ref{['thm:all_3_dicritical_semi_complete']}
  12. Preliminaries for the code
  13. The proof of Lemma \ref{['lemma:T1_T2_T3_T4']}
  14. The proof of Lemma \ref{['plusminus']}
  15. The proof of Lemma \ref{['fsub']}
  16. ...and 1 more sections

Key Result

Theorem 1

Let $n$ and $k$ be two integers such that $n>k\geq 4$. If $G$ is a $k$-critical graph on $n$ vertices, then $m(G)\geq \left\lceil\frac{(k+1)(k-2)n-k(k-3)}{2(k-1)}\right\rceil$. In other words, $g_k(n)\geq \left\lceil\frac{(k+1)(k-2)n-k(k-3)}{2(k-1)}\right\rceil$.

Figures (7)

  • Figure 1: The $3$-dicritical semi-complete digraphs, namely the bidirected complete graph $\overleftrightarrow{K_3}$, the directed wheel $\overrightarrow{W_3}$, the digraph $\mathcal{H}_5$, the rotative digraphs $\mathcal{R}(H_1,H_2)$ for every $H_1,H_2\in \{\overleftrightarrow{K_2}, \overrightarrow{C_3}\}$ and the Paley tournament on seven vertices $\mathcal{P}_7$. A big arrow linking two sets of vertices indicates that there is exactly one arc from every vertex in the first set to every vertex in the second set.
  • Figure 2: The oriented graph $O_5$.
  • Figure 3: A listing of all possible configurations for $i,j \in I$ with $i<j$. For the sake of better readability, the arcs in $A(C_i)\cup A(C_j)$ and the arcs from $V(C_j)$ to $V(C_i)$ are solid, and the arcs from $V(C_i)$ to $V(C_j)$ are dashed.
  • Figure 4: The digraph $O_4$.
  • Figure 5: The bidirected star on 4 vertices $\overleftrightarrow{S_4}$.
  • ...and 2 more figures

Theorems & Definitions (55)

  • Theorem 1: Kostochka and Yancey kostochkaJCTB109
  • Theorem 2: Toft toftSSMH5
  • Theorem 3: Stiebitz stiebitzComb7
  • Theorem 4: Luo, Ma, and Yang luoCPC32
  • Conjecture 5: Kostochka and Stiebitz kostochkaGC36
  • Conjecture 6: Kostochka and Stiebitz kostochkaGC36
  • Conjecture 7: Kostochka and Stiebitz kostochkaGC36
  • Conjecture 8: Hoshino and Kawarabayashi hoshinoComb35
  • Theorem 9
  • Theorem 10
  • ...and 45 more