Directed Hamiltonicity in Generalized Kneser Graphs

TL;DR

This paper proves the existence of directed Hamiltonian cycles and determines the dichromatic number in generalized Kneser graphs for certain parameters, extending understanding of their structural properties.

Contribution

It establishes directed Hamiltonicity and exact dichromatic number for generalized Kneser graphs, using a novel adaptation of the class graph framework.

Findings

Existence of directed Hamiltonian cycle for all s ≥ 3 and n > sk.

Dichromatic number of the oriented graph is exactly k.

Results apply to s-stable Kneser graphs, resolving their Hamiltonicity and dichromatic number.

Abstract

We prove that the canonical orientation of the generalized Kneser graph $KG(n,k,s)$ contains a directed Hamiltonian cycle for all integers $s \geq 3$ and $n>sk$. Furthermore, we establish that the dichromatic number of this oriented graph is exactly $k$. As a special case, our results apply to the $s$-stable Kneser graphs $K_{s\text{-stab}}(n,k)$, resolving their directed Hamiltonicity and dichromatic number. Our proof adapts the class graph framework of Ledezma and Pastine to the directed setting, leveraging cyclic rotations and friend class adjacencies to construct a single directed cycle spanning all vertices. This work provides a unified and strengthened perspective on the Hamiltonian properties of Kneser-type graphs.