Four Blocks Cycles C(k,1,1,1) in Digraphs

TL;DR

This paper improves the upper bound on the chromatic number of certain digraphs that do not contain subdivisions of a specific four-block cycle, reducing it from an exponential to a linear function of k.

Contribution

The paper provides a tighter bound of 18k on the chromatic number for digraphs lacking subdivisions of C(k,1,1,1), refining previous exponential bounds.

Findings

Chromatic number bound improved to 18k

No subdivisions of C(k,1,1,1) imply lower chromatic complexity

Enhanced understanding of cycle structures in digraphs

Abstract

A four blocks cycle C(k1,k2,k3,k4) is an oriented cycle formed by the union of four internally disjoint directed paths of lengths k1,k2,k3 and k4 respectively. El Mniny proved that if D is a digraph having a spanning out-tree T with no subdivisions of C(k, 1, 1, 1), then the chromatic number of D is at most 8^{3}k. In this paper, we will improve this bound to 18k.