arXiv subjects

Ole Gabsdil

Publications and source records attributed to Ole Gabsdil.

3 recordsLinked to original sources

arXiv2026

On sparse universally rigid graphs in $\mathbb{R}$

A graph is said to be 1-universally rigid if, for any generic realization of its vertices on the real line, the realization cannot be deformed out of the line in any higher-dimensional Euclidean space while preserving all edge lengths. In this paper, we investigate the sparsity of 1-universally rigid graphs in different manners. First, we show that an Erdős-Rényi random graph $G(n,p)$ is 1-universally rigid with high probability whenever $p \geq C\log n \cdot \log\log n/n$ for some sufficiently large constant $C>0$. These techniques also imply the existence of 1-universally rigid graphs of arbitrarily large girth, which resolves a problem suggested by Dantas e Moura, Jordán, and Silverman in a non-constructive manner. We also study how sparse a 1-universally rigid graph can be, proving that there exists an absolute constant $\varepsilon>0$ such that every sufficiently large 1-universally rigid graph on $n$ vertices has at least $(3/2+\varepsilon)n$ edges. This breaks the natural $3/2$ barrier arising from degree-2 extensions and makes progress towards a question of Jordán and Nguyen.

math.CO↗

arXiv2026

Supersaturation of induced even cycles in locally sparse graphs

A graph $Γ$ is $(c,t)$-sparse for $c > 0$ and $t \ge 1$ if for every pair of vertex subsets $A, B \subseteq V(Γ)$ with $|A|, |B| \ge t$, the number of edges $e(A,B)$ between them satisfies $ e(A,B) \le (1 - c)|A||B|$. In this paper, we prove that for every integer $\ell\ge2$, there are $\varepsilon > 0, C, C' > 0$ such that if an $n$-vertex graph $Γ$ is $(1-\varepsilon,t)$-sparse for some $t$, and has at least $Ct^{1-1/\ell}n^{1+1/\ell}$ edges, then $Γ$ contains at least $C'n^2t^{2\ell-2}$ induced copies of $C_{2\ell}$. This partially resolves a problem of Ding, Gao, Liu, Luan, and Sun.

math.CO↗

arXiv2025

On the Universality of Round Elimination Fixed Points

Recent work on distributed graph algorithms [e.g. STOC 2022, ITCS 2022, PODC 2020] has drawn attention to the following open question: are round elimination fixed points a universal technique for proving lower bounds? That is, given a locally checkable problem $Π$ that requires at least $Ω(\log n)$ rounds in the deterministic LOCAL model, can we always find a relaxation $Π'$ of $Π$ that is a nontrivial fixed point for the round elimination technique [see STOC 2016, PODC 2019]? If yes, then a key part of distributed computational complexity would be also decidable. The key obstacle so far has been a certain family of homomorphism problems [ITCS 2022], which require $Ω(\log n)$ rounds, but the only known proof is based on Marks' technique [J. AMS 2016]. We develop a new technique for constructing round elimination lower bounds systematically. Using so-called tripotent inputs we show that the aforementioned homomorphism problems indeed admit a lower bound proof that is based on round elimination fixed points. Hence we eliminate the only known obstacle for the universality of round elimination. Yet we also present a new obstacle: we show that there are some problems with inputs that require $Ω(\log n)$ rounds, yet there is no proof that is based on relaxations to nontrivial round elimination fixed points. Hence round elimination cannot be a universal technique for problems with inputs (but it might be universal for problems without inputs). We also prove the first fully general lower bound theorem that is applicable to any problem, with or without inputs, that is a fixed point in round elimination. Prior results of this form were only able to handle certain very restricted inputs.

cs.DC↗