Powers of Hamiltonian cycles in multipartite graphs

TL;DR

This paper establishes conditions under which a k-partite graph contains the (r-1)-st power of a Hamiltonian cycle, based on part sizes and vertex adjacency proportions.

Contribution

It proves a new minimum degree condition ensuring the existence of the (r-1)-st power of a Hamiltonian cycle in multipartite graphs.

Findings

Conditions guarantee Hamiltonian cycle powers in multipartite graphs

Part size constraints are critical for cycle powers

Vertex adjacency thresholds are sufficient for cycle embedding

Abstract

We prove that if $G$ is a $k$-partite graph on $n$ vertices in which all of the parts have order at most $n/r$ and every vertex is adjacent to at least a $1-1/r+o(1)$ proportion of the vertices in every other part, then $G$ contains the $(r-1)$-st power of a Hamiltonian cycle