Directed Hamiltonicity in Generalized Kneser Graphs

TL;DR

This paper investigates directed coloring parameters for generalized Kneser graphs $KG(n,k,s)$ and Johnson digraphs, focusing on $\chi_{\vec{}}$ and $\chi_{\vec{\ell}}$. It yields tight asymptotics: $\chi_{\vec{}}(KG(n,k,s))=\Theta(n-2k+s+1)$ for fixed $s$ and $2\le k\le n/2$, and $\chi_{\vec{\ell}}(KG(n,k,s))=\Theta(n\ln n)$ for fixed $s$ with $k\le n^{1/2-\varepsilon}$, using a subgraph reduction to $KG(n-s,k-s)$ and an explicit orientation. The Johnson digraphs are examined with a conjectured bound $\Theta(n/k)$ for the dichromatic number and partial progress. The work blends the class-graph framework of Ledezma and Pastine, topological tools such as the Lusternik–Schnirelmann–Borsuk theorem, and probabilistic concentration to unify and extend directed coloring properties of Kneser-type graphs, and outlines open problems on graph products and Johnson digraphs for future research.

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.

Theorems & Definitions (15)

• Definition 1: Generalized Kneser Graph $KG(n, k, s)$
• Definition 2: Johnson Digraph $J(n, k)$
• Theorem 1: harutyunyan2025colouring, Theorem 6
• Theorem 2: harutyunyan2025colouring, Theorem 19
• Theorem 3: alon1992choice, Theorem 1
• Theorem 4
• proof
• Theorem 5
• proof
• Lemma 6