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

TL;DR

This paper proves Cohen et al.'s conjecture for six-block cycles in strongly connected digraphs, establishing bounds on chromatic number when such cycles are absent, and improves bounds for specific cases.

Contribution

It confirms the conjecture for all six-block cycles C(k,1,1,1,1,1), providing bounds on chromatic number and reducing these bounds for k=1.

Findings

Chromatic number is at most O(k) for cycles C(k,1,1,1,1,1) in strongly connected digraphs.

Confirmed Cohen et al.'s conjecture for a new class of six-block cycles.

Reduced bounds on chromatic number when k=1.

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.