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Minimum acyclic number and maximum dichromatic number of oriented triangle-free graphs of a given order

Pierre Aboulker, Frédéric Havet, François Pirot, Juliette Schabanel

Abstract

Let $D$ be a digraph. Its acyclic number $\vecα(D)$ is the maximum order of an acyclic induced subdigraph and its dichromatic number $\vecχ(D)$ is the least integer $k$ such that $V(D)$ can be partitioned into $k$ subsets inducing acyclic subdigraphs. We study ${\vec a}(n)$ and $\vec t(n)$ which are the minimum of $\vecα(D)$ and the maximum of $\vecχ(D)$, respectively, over all oriented triangle-free graphs of order $n$. For every $ε>0$ and $n$ large enough, we show $(1/\sqrt{2} - ε) \sqrt{n\log n} \leq \vec{a}(n) \leq \frac{107}{8} \sqrt n \log n$ and $\frac{8}{107} \sqrt n/\log n \leq \vec{t}(n) \leq (\sqrt 2 + ε) \sqrt{n/\log n}$. We also construct an oriented triangle-free graph on 25 vertices with dichromatic number~3, and show that every oriented triangle-free graph of order at most 17 has dichromatic number at most 2.

Minimum acyclic number and maximum dichromatic number of oriented triangle-free graphs of a given order

Abstract

Let be a digraph. Its acyclic number is the maximum order of an acyclic induced subdigraph and its dichromatic number is the least integer such that can be partitioned into subsets inducing acyclic subdigraphs. We study and which are the minimum of and the maximum of , respectively, over all oriented triangle-free graphs of order . For every and large enough, we show and . We also construct an oriented triangle-free graph on 25 vertices with dichromatic number~3, and show that every oriented triangle-free graph of order at most 17 has dichromatic number at most 2.
Paper Structure (12 sections, 24 theorems, 30 equations, 6 figures)

This paper contains 12 sections, 24 theorems, 30 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Probabilistic tools
  4. A useful orientation
  5. Bounds on $\vec{a}(n)$ and $\vec{t}(n)$
  6. Upper bound on $\vec{a}(n)$
  7. Upper bound on $\vec{t}(n)$
  8. A deterministic approach
  9. Linear forests
  10. The construction
  11. Getting a better bound in the case k=2
  12. Lower bound on the number of vertices of oriented triangle-free graphs with dichromatic number 3

Key Result

Lemma 2.1

Let $X$ be a random variable distributed according to the binomial distribution $\mathcal{B}(n,p)$, for some integer $n$ and some $p\in (0,1)$. For every $x\in (0,1)$, we let $\Lambda^*(x) \coloneqq x \ln \frac{x}{p} + (1-x) \ln \frac{1-x}{1-p}$. Then, for every integer $k < n$, one has

Figures (6)

  • Figure 1: A 2-dicoloration of the 6-backward-blowup of the two orientations of $C_5$ that contain no directed path of length 4.
  • Figure 2: The digraph $D_{25}$. Left (only the forward arcs are represented): in green a pair of matched vertices, and in orange a pack. Right: the three main types of directed cycles.
  • Figure 3: Left: a red backward cycle, when $U_1$ has five red vertices and $U_2$ and $U_5$ at least one. Right: The case with a pack with a $4$-$1$ colour partition.
  • Figure 4: 2-dicolouring of $D_{25}$ minus a vertex (left), $D_{25}$ minus a forward arc (dotted blue) (center), $D_{25}$ minus a backward arc (dotted red) (right).
  • Figure 5: The case where $G$ contains a $C_7$
  • ...and 1 more figures

Theorems & Definitions (41)

  • Conjecture 1: Harutyunyan and McDiarmid HCunpublished
  • Conjecture 2: Alon, Pachs, Solymosi APS01
  • Conjecture 3
  • Conjecture 4
  • Lemma 2.1: c.f. KaMc10
  • Lemma 2.2: Chernoff's bound
  • Lemma 2.3: Lovász Local Lemma
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • ...and 31 more