Powers of Hamilton cycles in dense graphs perturbed by a random geometric graph

TL;DR

This paper determines the conditions under which a dense graph combined with a random geometric graph almost surely contains the k-th power of a Hamilton cycle, advancing understanding of spanning structures in perturbed graphs.

Contribution

It provides asymptotically optimal conditions on the radius r for the union of a dense graph and a random geometric graph to contain the k-th power of a Hamilton cycle.

Findings

Identifies conditions on r for Hamilton cycle powers in perturbed graphs

Establishes asymptotically optimal thresholds for various parameters

Extends results to applications like F-factors

Abstract

Let $G$ be a graph obtained as the union of some $n$-vertex graph $H_n$ with minimum degree $\delta(H_n)\geq\alpha n$ and a $d$-dimensional random geometric graph $G^d(n,r)$. We investigate under which conditions for $r$ the graph $G$ will a.a.s. contain the $k$-th power of a Hamilton cycle, for any choice of $H_n$. We provide asymptotically optimal conditions for $r$ for all values of $\alpha$, $d$ and $k$. This has applications in the containment of other spanning structures, such as $F$-factors.