## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Thursday, December 21, 2023, 12:15 pm

Duration: 30 minutes

Location: CAB G51

Speaker: Kalina Petrova

## On Connectivity in Random Graph Models with Limited Dependencies

For any positive edge density $p$, a random graph in the Erdös-Rényi $G(n,p)$ model is connected with non-zero probability, since all edges are mutually independent. We consider random graph models in which edges that do not share endpoints are independent while incident edges may be dependent and ask: what is the minimum probability $\rho(n)$, such that for any distribution $D$ (in this model) on graphs with $n$ vertices in which each potential edge has a marginal probability of being present at least $\rho(n)$, a graph drawn from $D$ is connected with non-zero probability? As it turns out, the condition "edges that do not share endpoints are independent" needs to be clarified and the answer to the question above is sensitive to the specification. In fact, we formalize this intuitive description into a strict hierarchy of five independence conditions, which we show to have at least three different behaviors for the threshold $\rho(n)$. For each condition, we provide upper and lower bounds for $\rho(n)$. In the strongest condition, the *coloring model* (which includes, e.g., random geometric graphs), we prove that $\rho(n) → 2 - \phi ≈ 0.38$ for $n \rightarrow \infty$. This separates it from the weaker independence conditions we consider, as there we prove that $\rho(n) > 0.5 - o(n)$. In stark contrast to the coloring model, for our weakest independence condition — pairwise independence of non-adjacent edges — we show that $\rho(n)$ lies within $O(1/n^2)$ of the threshold $1-2/n$ for completely arbitrary distributions. This is joint work with Johannes Lengler, Anders Martinsson, Patrick Schnider, Raphael Steiner, Simon Weber, and Emo Welzl.

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