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

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.
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Taxonomy
TopicsAdvanced Graph Theory Research · graph theory and CDMA systems · Graph Labeling and Dimension Problems
