About subdivisions of four blocks cycles $C(k_1,1,k_3,1)$ in digraphs with large chromatic number

TL;DR

This paper proves a conjecture about the chromatic number bounds of certain strongly connected digraphs avoiding specific subdivisions of four-block cycles, especially when two of the block lengths are one, with implications for Hamiltonian digraphs.

Contribution

It confirms Cohen et al.'s conjecture for cycles with two blocks of length one, establishing an O((k1+k3)^2) bound, and provides a tighter bound for Hamiltonian digraphs.

Findings

Confirmed the conjecture for cycles with k2=k4=1.

Established a quadratic bound on the chromatic number.

Provided a linear bound for Hamiltonian digraphs.

Abstract

A cycle with four blocks $C(k_{1}, k_{2},k_{3},k_{4})$ is an oriented cycle formed of four blocks of lengths $k_{1}, k_{2}, k_{3}$ and $k_{4}$ respectively. Recently, Cohen et al. conjectured that for every positive integers $k_{1}, k_{2}, k_{3}, k_{4}$, there is an integer $g(k_{1},k_{2},k_{3},k_{4})$ such that every strongly connected digraph $D$ containing no subdivisions of $C(k_{1},k_{2},k_{3},k_{4})$ has a chromatic number at most $g(k_{1},k_{2},k_{3},k_{4})$. This conjecture is confirmed by Cohen et al. for the case of $C(1,1,1,1)$ and by Al-Mniny for the case of $C(k_1,1,1,1)$. In this paper, we affirm Cohen et al.'s conjecture for the case where $k_2=k_4=1$, namely $g(k_1,1,k_3,1) =O({(k_1+k_3)}^2)$. Moreover, we show that if in addition $D$ is Hamiltonian, then the chromatic number of $D$ is at most $6k$, with $k=\textrm{max}\{k_1,k_3\}.$