Work

On the Local Resilience of Random Geometric Graphs With Respect to Connectivity and Long Cycles

Year: 2026

Type: article

Abstract: ABSTRACT Given an increasing graph property , a graph is ‐resilient with respect to if, for every spanning subgraph where each vertex keeps more than a ‐proportion of its neighbors, has property . We study the above notion of local resilience when is a random geometric graph obtained by embedding vertices independently and uniformly at random in , and connecting two vertices by an edge if the distance between them is at most . First, we focus on connectivity. We show that, for every , when is a constant factor above the sharp threshold for connectivity of , the random geometric graph is ‐resilient for the property of being ‐connected, with of the same order as the expected degree. However, contrary to binomial random graphs, for sufficiently small , connectivity is not born ‐resilient in 2‐dimensional random geometric graphs. Second, we study local resilience with respect to the property of containing long cycles. We show that, when is a constant factor above , is ‐resilient with respect to containing cycles of all lengths between constant and . Proving ‐resilience for Hamiltonicity remains elusive with our techniques. Nevertheless, we show that is ‐resilient with respect to Hamiltonicity for a fixed constant .

Source: Random Structures and Algorithms

DOI: 10.1002/rsa.70100

Authors: Alberto Espuny Díaz, Lyuben Lichev, Alexandra Wesolek

Institutions: Universitat de Barcelona, TU Wien, Centre National de la Recherche Scientifique, Université de Bordeaux, Laboratoire Bordelais de Recherche en Informatique

Language: English

Topic: Limits and Structures in Graph Theory

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $4,550

Funders: Deutsche Forschungsgemeinschaft, Austrian Science Fund