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Subdivisions of Six-Blocks Cycles C(k,1,1,1,1,1) in Strong Digraphs

Hiba Ayoub, Soukaina Zayat, Darine Al-Mniny

TL;DR

This work resolves Cohen et al.'s conjecture for six-block cycles by decomposing any strong digraph $D$ avoiding subdivisions of $C(k,1,1,1,1,1)$ into layer-based components $D_i^1$, $D_i^2$, and $D_i^3$ via a final out-tree, then bounding each component's chromatic number using antidirected-cycle theorems and degeneracy arguments. The authors show $oxed{\chi(D_i^1) \le 24}$, $oxed{\chi(D_i^2) \le 7}$, and $oxed{\chi(D_i^3) \le 24}$, which together yield $oxed{\chi(D) \le 7\cdot 24^2\cdot k}$; they also provide refinements for the $k=1$ case, achieving $oxed{\chi(D) \le 7\cdot 16^2\cdot k}$. The results deliver a linear-in-$k$ bound on the chromatic number for six-block subdivisions, advancing the broader program of Cohen et al. and enriching structural tools for analyzing subdivision-free strong digraphs. The methods highlight how a layered, tree-based decomposition constrains complex directed cycles and enables global coloring bounds with explicit constants.

Abstract

A cycle C(k1,k2,...,kn) is the oriented cycle formed of n blocks of lengths k1,k2,...,kn-1 and kn respectively. In 2018 Cohen et al. conjectured that for every positive integers k1,k2,...,kn there exists a constant g(k1,k2,...,kn) such that every strongly connected digraph containing no subdivisions of C(k1,k2,...,kn) has a chromatic number at most g(k1,k2,...,kn). In their paper, Cohen et al. confirmed the conjecture for cycles with two blocks and for cycles with four blocks having all its blocks of length 1. Recently, the conjecture was proved for special types of four-blocks cycles. In this paper, we confirm Cohen et al.'s conjecture for all six-blocks cycles C(k,1,1,1,1,1). Precisely, for any integer k, we prove that every strongly connected digraph containing no subdivisions of C(k,1,1,1,1,1) has a chromatic number at most O(k), and we significantly reduce the chromatic number in case k=1.

Subdivisions of Six-Blocks Cycles C(k,1,1,1,1,1) in Strong Digraphs

TL;DR

This work resolves Cohen et al.'s conjecture for six-block cycles by decomposing any strong digraph avoiding subdivisions of into layer-based components , , and via a final out-tree, then bounding each component's chromatic number using antidirected-cycle theorems and degeneracy arguments. The authors show , , and , which together yield ; they also provide refinements for the case, achieving . The results deliver a linear-in- bound on the chromatic number for six-block subdivisions, advancing the broader program of Cohen et al. and enriching structural tools for analyzing subdivision-free strong digraphs. The methods highlight how a layered, tree-based decomposition constrains complex directed cycles and enables global coloring bounds with explicit constants.

Abstract

A cycle C(k1,k2,...,kn) is the oriented cycle formed of n blocks of lengths k1,k2,...,kn-1 and kn respectively. In 2018 Cohen et al. conjectured that for every positive integers k1,k2,...,kn there exists a constant g(k1,k2,...,kn) such that every strongly connected digraph containing no subdivisions of C(k1,k2,...,kn) has a chromatic number at most g(k1,k2,...,kn). In their paper, Cohen et al. confirmed the conjecture for cycles with two blocks and for cycles with four blocks having all its blocks of length 1. Recently, the conjecture was proved for special types of four-blocks cycles. In this paper, we confirm Cohen et al.'s conjecture for all six-blocks cycles C(k,1,1,1,1,1). Precisely, for any integer k, we prove that every strongly connected digraph containing no subdivisions of C(k,1,1,1,1,1) has a chromatic number at most O(k), and we significantly reduce the chromatic number in case k=1.

Paper Structure

This paper contains 6 sections, 18 theorems, 3 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The existence of $S$-$C(k,1,1,1,1,1)$ in strong digraphs
  4. Coloring $D_i^1$ and $D_i^2$
  5. The landscape of antidirected cycles in $D_{i}^{3}$ and coloring $D_i^3$
  6. Main Theorem

Key Result

Theorem 1.1

(Cohen et al. hjk) For any positive integers $b,c$, there exists an acyclic digraph $D$ with $\chi(D) \geq c$ in which all oriented cycles have more than $b$ blocks.

Figures (2)

  • Figure 1: All possible positions of $z_1, z_{s-1}$, and $z_s$ on the tree $T$.
  • Figure 2: Illustration of Assertion \ref{['assa']} and Assertion \ref{['ass1']}.

Theorems & Definitions (38)

  • Conjecture 1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • Theorem 3.1
  • ...and 28 more