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Generalized Turán problem for directed cycles

Andrzej Grzesik, Justyna Jaworska, Bartłomiej Kielak, Piotr Kuc, Tomasz Ślusarczyk

TL;DR

This work advances the generalized Turán problem for directed cycles by determining the order of growth of $ex(n, \overrightarrow{C_k}, \overrightarrow{C_\ell})$ for all $k,\ell \ge 3$, showing a dichotomy between $\Theta(n^k)$ and $\Theta(n^{k-1})$ depending on divisibility. In the dense regime, the leading term often comes from a balanced blow-up of $\overrightarrow{C_d}$, with $d$ the smallest divisor $>2$ of $k$ not dividing $\ell$, yielding $ex(n) = \frac{n}{k}(\frac{n}{d})^{k-1} + o(n^k)$ under suitable conditions; when these fail, a random bipartite orientation furnishes the extremal bound $ex(n) = \frac{2}{k}(\frac{n}{4})^k + o(n^k)$. The paper also analyzes small cycles ($k=3,4,5$) in depth, providing exact or near-exact results in several cases and highlighting diverse extremal constructions, including iterated blow-ups and more intricate configurations. Additionally, sparse-case results, extensions to directed graphs, and a discussion of cycle-type generalizations broaden the scope and open questions for future work.

Abstract

For integers $k, \ell \geq 3$, let $\mathrm{ex}(n, \overrightarrow{C_k}, \overrightarrow{C_\ell})$ denote the maximum number of directed cycles of length $k$ in any oriented graph on $n$ vertices which does not contain a directed cycle of length $\ell$. We establish the order of magnitude of $\mathrm{ex}(n, \overrightarrow{C_k}, \overrightarrow{C_\ell})$ for every $k$ and $\ell$ and determine its value up to a lower error term when $k \nmid \ell$ and $\ell$ is large enough. Additionally, we calculate the value of $\mathrm{ex}(n, \overrightarrow{C_k}, \overrightarrow{C_\ell})$ for some other specific pairs $(k, \ell)$ showing that a diverse class of extremal constructions can appear for small values of $\ell$.

Generalized Turán problem for directed cycles

TL;DR

This work advances the generalized Turán problem for directed cycles by determining the order of growth of for all , showing a dichotomy between and depending on divisibility. In the dense regime, the leading term often comes from a balanced blow-up of , with the smallest divisor of not dividing , yielding under suitable conditions; when these fail, a random bipartite orientation furnishes the extremal bound . The paper also analyzes small cycles () in depth, providing exact or near-exact results in several cases and highlighting diverse extremal constructions, including iterated blow-ups and more intricate configurations. Additionally, sparse-case results, extensions to directed graphs, and a discussion of cycle-type generalizations broaden the scope and open questions for future work.

Abstract

For integers , let denote the maximum number of directed cycles of length in any oriented graph on vertices which does not contain a directed cycle of length . We establish the order of magnitude of for every and and determine its value up to a lower error term when and is large enough. Additionally, we calculate the value of for some other specific pairs showing that a diverse class of extremal constructions can appear for small values of .

Paper Structure

This paper contains 20 sections, 18 theorems, 37 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Order of magnitude
  4. The dense setting
  5. The cycle blow-up case
  6. Assumption $2\nmid k$ or $2\mid \ell$.
  7. Assumption $d = 3$.
  8. Assumption $d = 4$.
  9. The random bipartite case
  10. Proof of Lemma \ref{['lem:bipartite_optimal']}
  11. Proof of Lemma \ref{['lem:knn_rand_optimal']}
  12. Small cycles
  13. Maximizing $\overrightarrow{C\space}_{\space3}$
  14. Maximizing $\overrightarrow{C\space}_{\space4}$
  15. Cases $\ell = 5$ and $\ell = 7$.
  16. ...and 5 more sections

Key Result

Theorem 2

Let $a_1, a_2, \ldots, a_k$ be positive integers that satisfy $\gcd (a_1, a_2, \ldots, a_k) = 1$. Denote $d_i = \gcd (a_1, \ldots, a_i)$. Then any integer can be represented as $x_1a_1 + x_2a_2 + \ldots + x_ka_k$ for non-negative integers $x_1, \ldots, x_k$.

Figures (10)

  • Figure 1: Constructions providing lower bounds for Theorem \ref{['thm:order_of_magnitude']}.
  • Figure 2: Examples for $k=4$ and $t=3$, $t=4$ respectively.
  • Figure 3: For $k = 4$, existence of $\overrightarrow{T_{3}}$ implies existence of homomorphic images of $\overrightarrow{C\space}_{\space5}$ and $\overrightarrow{C\space}_{\space7}$.
  • Figure 4: Example for $\ell = 8$, $m = 1$
  • Figure 5: An iterated blow-up of $\overrightarrow{C\space}_{\space4}$.
  • ...and 5 more figures

Theorems & Definitions (59)

  • Theorem 2: Brauer brauer1942problem
  • Corollary 3
  • proof
  • Theorem 4
  • proof
  • Definition 5
  • proof
  • proof
  • Theorem 8
  • Lemma 9
  • ...and 49 more