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1. Large monochromatic components in hypergraphs with large minimum codegree

   https://doi.org/10.1002/jgt.23044  

   Bal, Deepak; DeBiasio, Louis (October 2023, Journal of Graph Theory)

   Abstract A result of Gyárfás says that for every 3‐coloring of the edges of the complete graph , there is a monochromatic component of order at least , and this is best possible when 4 divides . Furthermore, for all and every ‐coloring of the edges of the complete ‐uniform hypergraph , there is a monochromatic component of order at least and this is best possible for all . Recently, Guggiari and Scott and independently Rahimi proved a strengthening of the graph case in the result above which says that the same conclusion holds if is replaced by any graph on vertices with minimum degree at least ; furthermore, this bound on the minimum degree is best possible. We prove a strengthening of the case in the result above which says that the same conclusion holds if is replaced by any ‐uniform hypergraph on vertices with minimum ‐degree at least ; furthermore, this bound on the ‐degree is best possible. 

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2. Exact Community Recovery in Correlated Stochastic Block Models

   Gaudio, Julia; Racz, Miklos Z.; Sridhar, Anirudh (July 2022, Proceedings of Machine Learning Research)

   We consider the problem of learning latent community structure from multiple correlated networks. We study edge-correlated stochastic block models with two balanced communities, focusing on the regime where the average degree is logarithmic in the number of vertices. Our main result derives the precise information-theoretic threshold for exact community recovery using multiple correlated graphs. This threshold captures the interplay between the community recovery and graph matching tasks. In particular, we uncover and characterize a region of the parameter space where exact community recovery is possible using multiple correlated graphs, even though (1) this is information-theoretically impossible using a single graph and (2) exact graph matching is also information-theoretically impossible. In this regime, we develop a novel algorithm that carefully synthesizes algorithms from the community recovery and graph matching literatures. 

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3. Seeded graph matching via large neighborhood statistics

   https://doi.org/10.1002/rsa.20934  

   Mossel, Elchanan; Xu, Jiaming (June 2020, Random Structures & Algorithms)

   We study a noisy graph isomorphism problem, where the goal is to perfectly recover the vertex correspondence between two edge‐correlated graphs, with an initial seed set of correctly matched vertex pairs revealed as side information. We show that it is possible to achieve the information‐theoretic limit of graph sparsity in time polynomial in the number of verticesn. Moreover, we show the number of seeds needed for perfect recovery in polynomial‐time can be as low asin the sparse graph regime (with the average degree smaller than) andin the dense graph regime, for a small positive constant. Unlike previous work on graph matching, which used small neighborhoods or small subgraphs with a logarithmic number of vertices in order to match vertices, our algorithms match vertices if their large neighborhoods have a significant overlap in the number of seeds. 

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4. Acyclic graphs with at least 2ℓ + 1 vertices are ℓ‐recognizable

   https://doi.org/10.1002/jgt.23027  

   Kostochka, Alexandr V.; Nahvi, Mina; West, Douglas B.; Zirlin, Dara (August 2023, Journal of Graph Theory)

   Abstract The ‐deckof an ‐vertex graph is the multiset of subgraphs obtained from it by deleting vertices. A family of ‐vertex graphs is ‐recognizableif every graph having the same ‐deck as a graph in the family is also in the family. We prove that the family of ‐vertex graphs with no cycles is ‐recognizable when (except for ). As a consequence, the family of ‐vertex trees is ‐recognizable when and . It is known that this fails when . 

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5. Recovering a hidden community beyond the Kesten–Stigum threshold in O(|E|log*|V|) time

   https://doi.org/10.1017/jpr.2018.22  

   Hajek, Bruce; Wu, Yihong; Xu, Jiaming (June 2018, Journal of Applied Probability)

   Abstract Community detection is considered for a stochastic block model graph of n vertices, with K vertices in the planted community, edge probability p for pairs of vertices both in the community, and edge probability q for other pairs of vertices. The main focus of the paper is on weak recovery of the community based on the graph G , with o ( K ) misclassified vertices on average, in the sublinear regime n 1- o (1) ≤ K ≤ o ( n ). A critical parameter is the effective signal-to-noise ratio λ = K 2 ( p - q ) 2 / (( n - K ) q ), with λ = 1 corresponding to the Kesten–Stigum threshold. We show that a belief propagation (BP) algorithm achieves weak recovery if λ > 1 / e, beyond the Kesten–Stigum threshold by a factor of 1 / e. The BP algorithm only needs to run for log * n + O (1) iterations, with the total time complexity O (| E |log * n ), where log * n is the iterated logarithm of n . Conversely, if λ ≤ 1 / e, no local algorithm can asymptotically outperform trivial random guessing. Furthermore, a linear message-passing algorithm that corresponds to applying a power iteration to the nonbacktracking matrix of the graph is shown to attain weak recovery if and only if λ > 1. In addition, the BP algorithm can be combined with a linear-time voting procedure to achieve the information limit of exact recovery (correctly classify all vertices with high probability) for all K ≥ ( n / log n ) (ρ BP + o (1)), where ρ BP is a function of p / q . 

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