On the local resilience of random geometric graphs with respect to connectivity and long cycles
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2024
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| author | Díaz, Alberto Espuny Lichev, Lyuben Wesolek, Alexandra |
| author_facet | Díaz, Alberto Espuny Lichev, Lyuben Wesolek, Alexandra |
| contents | Given an increasing graph property $\mathcal{P}$, a graph $G$ is $α$-resilient with respect to $\mathcal{P}$ if, for every spanning subgraph $H\subseteq G$ where each vertex keeps more than a $(1-α)$-proportion of its neighbours, $H$ has property $\mathcal{P}$. We study the above notion of local resilience with $G$ being a random geometric graph $G_d(n,r)$ obtained by embedding $n$ vertices independently and uniformly at random in $[0,1]^d$, and connecting two vertices by an edge if the distance between them is at most $r$.
First, we focus on connectivity. We show that, for every $\varepsilon>0$, for $r$ a constant factor above the sharp threshold for connectivity $r_c$ of $G_d(n,r)$, the random geometric graph is $(1/2-\varepsilon)$-resilient for the property of being $k$-connected, with $k$ of the same order as the expected degree. However, contrary to binomial random graphs, for sufficiently small $\varepsilon>0$, connectivity is not born $(1/2-\varepsilon)$-resilient in $2$-dimensional random geometric graphs.
Second, we study local resilience with respect to the property of containing long cycles. We show that, for $r$ a constant factor above $r_c$, $G_d(n,r)$ is $(1/2-\varepsilon)$-resilient with respect to containing cycles of all lengths between constant and $2n/3$. Proving $(1/2-\varepsilon)$-resilience for Hamiltonicity remains elusive with our techniques. Nevertheless, we show that $G_d(n,r)$ is $α$-resilient with respect to Hamiltonicity for a fixed constant $α= α(d)<1/2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_09921 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the local resilience of random geometric graphs with respect to connectivity and long cycles Díaz, Alberto Espuny Lichev, Lyuben Wesolek, Alexandra Combinatorics Given an increasing graph property $\mathcal{P}$, a graph $G$ is $α$-resilient with respect to $\mathcal{P}$ if, for every spanning subgraph $H\subseteq G$ where each vertex keeps more than a $(1-α)$-proportion of its neighbours, $H$ has property $\mathcal{P}$. We study the above notion of local resilience with $G$ being a random geometric graph $G_d(n,r)$ obtained by embedding $n$ vertices independently and uniformly at random in $[0,1]^d$, and connecting two vertices by an edge if the distance between them is at most $r$. First, we focus on connectivity. We show that, for every $\varepsilon>0$, for $r$ a constant factor above the sharp threshold for connectivity $r_c$ of $G_d(n,r)$, the random geometric graph is $(1/2-\varepsilon)$-resilient for the property of being $k$-connected, with $k$ of the same order as the expected degree. However, contrary to binomial random graphs, for sufficiently small $\varepsilon>0$, connectivity is not born $(1/2-\varepsilon)$-resilient in $2$-dimensional random geometric graphs. Second, we study local resilience with respect to the property of containing long cycles. We show that, for $r$ a constant factor above $r_c$, $G_d(n,r)$ is $(1/2-\varepsilon)$-resilient with respect to containing cycles of all lengths between constant and $2n/3$. Proving $(1/2-\varepsilon)$-resilience for Hamiltonicity remains elusive with our techniques. Nevertheless, we show that $G_d(n,r)$ is $α$-resilient with respect to Hamiltonicity for a fixed constant $α= α(d)<1/2$. |
| title | On the local resilience of random geometric graphs with respect to connectivity and long cycles |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2406.09921 |