An Undecidability Result on Limits of Sparse Graphs

Endre Csóka

doi:10.37236/2340

An Undecidability Result on Limits of Sparse Graphs

The Electronic Journal of Combinatorics ◽

10.37236/2340 ◽

2012 ◽

Vol 19 (2) ◽

Author(s):

Endre Csóka

Keyword(s):

Random Graph ◽

Bounded Degree ◽

Sparse Graphs ◽

Random Node ◽

Bounded Degree Graphs ◽

Undecidability Result

Given a set $\mathcal{B}$ of finite rooted graphs and a radius $r$ as an input, we prove that it is undecidable to determine whether there exists a sequence $(G_i)$ of finite bounded degree graphs such that the rooted $r$-radius neighbourhood of a random node of $G_i$ is isomorphic to a rooted graph in $\mathcal{B}$ with probability tending to 1. Our proof implies a similar result for the case where the sequence $(G_i)$ is replaced by a unimodular random graph.

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Random Perturbation of Sparse Graphs

The Electronic Journal of Combinatorics ◽

10.37236/9510 ◽

2021 ◽

Vol 28 (2) ◽

Author(s):

Max Hahn-Klimroth ◽

Giulia Maesaka ◽

Yannick Mogge ◽

Samuel Mohr ◽

Olaf Parczyk

Keyword(s):

Random Graph ◽

Minimum Degree ◽

Random Perturbation ◽

Hamilton Cycle ◽

Hamilton Cycles ◽

Bounded Degree ◽

Sparse Graphs ◽

Fixed Constant ◽

Bounded Degree Trees ◽

Random Part

In the model of randomly perturbed graphs we consider the union of a deterministic graph $\mathcal{G}_\alpha$ with minimum degree $\alpha n$ and the binomial random graph $\mathbb{G}(n,p)$. This model was introduced by Bohman, Frieze, and Martin and for Hamilton cycles their result bridges the gap between Dirac's theorem and the results by Pósa and Korshunov on the threshold in $\mathbb{G}(n,p)$. In this note we extend this result in $\mathcal{G}_\alpha\cup\mathbb{G}(n,p)$ to sparser graphs with $\alpha=o(1)$. More precisely, for any $\varepsilon>0$ and $\alpha \colon \mathbb{N} \mapsto (0,1)$ we show that a.a.s. $\mathcal{G}_\alpha\cup \mathbb{G}(n,\beta /n)$ is Hamiltonian, where $\beta = -(6 + \varepsilon) \log(\alpha)$. If $\alpha>0$ is a fixed constant this gives the aforementioned result by Bohman, Frieze, and Martin and if $\alpha=O(1/n)$ the random part $\mathbb{G}(n,p)$ is sufficient for a Hamilton cycle. We also discuss embeddings of bounded degree trees and other spanning structures in this model, which lead to interesting questions on almost spanning embeddings into $\mathbb{G}(n,p)$.

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