More Like this

1. Immersions and Albertson’s Conjecture

   https://doi.org/10.4230/LIPIcs.SoCG.2025.50  

   Fox, Jacob; Pach, János; Suk, Andrew (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Aichholzer, Oswin; Wang, Haitao (Ed.)

   A graph is said to contain K_k (a clique of size k) as a weak immersion if it has k vertices, pairwise connected by edge-disjoint paths. In 1989, Lescure and Meyniel made the following conjecture related to Hadwiger’s conjecture: Every graph of chromatic number k contains K_k as a weak immersion. We prove this conjecture for graphs with at most 1.4(k-1) vertices. As an application, we make some progress on Albertson’s conjecture on crossing numbers of graphs, according to which every graph G with chromatic number k satisfies cr(G) ≥ cr(K_k). In particular, we show that the conjecture is true for all graphs of chromatic number k, provided that they have at most 1.4(k-1) vertices and k is sufficiently large. 

   more » « less
2. Finding a planted clique by adaptive probing

   https://doi.org/10.30757/ALEA.v17-30  

   Racz, Miklos Z; Schiffer, Benjamin (August 2020, Alea)

   null (Ed.)

   We consider a variant of the planted clique problem where we are allowed unbounded computational time but can only investigate a small part of the graph by adaptive edge queries. We determine (up to logarithmic factors) the number of queries necessary both for detecting the presence of a planted clique and for finding the planted clique. Specifically, let $$G \sim G(n,1/2,k)$$ be a random graph on $$n$$ vertices with a planted clique of size $$k$$. We show that no algorithm that makes at most $q = o(n^2 / k^2 + n)$ adaptive queries to the adjacency matrix of $$G$$ is likely to find the planted clique. On the other hand, when $$k \geq (2+\epsilon) \log_2 n$$ there exists a simple algorithm (with unbounded computational power) that finds the planted clique with high probability by making $$q = O( (n^2 / k^2) \log^2 n + n \log n)$$ adaptive queries. For detection, the additive $$n$$ term is not necessary: the number of queries needed to detect the presence of a planted clique is $n^2 / k^2$ (up to logarithmic factors). 

   more » « less
3. Sharp bounds for decomposing graphs into edges and triangles

   https://doi.org/10.1017/S0963548320000358  

   Blumenthal, Adam; Lidický, Bernard; Pehova, Yanitsa; Pfender, Florian; Pikhurko, Oleg; Volec, Jan (March 2021, Combinatorics, Probability and Computing)

   null (Ed.)

   Abstract For a real constant α , let $$\pi _3^\alpha (G)$$ be the minimum of twice the number of K 2 ’s plus α times the number of K 3 ’s over all edge decompositions of G into copies of K 2 and K 3 , where K r denotes the complete graph on r vertices. Let $$\pi _3^\alpha (n)$$ be the maximum of $$\pi _3^\alpha (G)$$ over all graphs G with n vertices. The extremal function $$\pi _3^3(n)$$ was first studied by Győri and Tuza ( Studia Sci. Math. Hungar. 22 (1987) 315–320). In recent progress on this problem, Král’, Lidický, Martins and Pehova ( Combin. Probab. Comput. 28 (2019) 465–472) proved via flag algebras that $$\pi _3^3(n) \le (1/2 + o(1)){n^2}$$ . We extend their result by determining the exact value of $$\pi _3^\alpha (n)$$ and the set of extremal graphs for all α and sufficiently large n . In particular, we show for α = 3 that K n and the complete bipartite graph $${K_{\lfloor n/2 \rfloor,\lceil n/2 \rceil }}$$ are the only possible extremal examples for large n . 

   more » « less
4. Learning Simplicial Complexes from Persistence Diagrams

   Belton, Robin Lynne; Fasy, Brittany Terese; Mertz, Rostik; Micka, Samuel; Millman, David L.; Salinas, Daniel; Schenfisch, Anna; Schupbach, Jordan; Williams, Lucia (August 2018, Canadian Conference on Computational Geometry)

   Topological Data Analysis (TDA) studies the “shape” of data. A common topological descriptor is the persistence diagram, which encodes topological features in a topological space at different scales. Turner, Mukherjee, and Boyer showed that one can reconstruct a simplicial complex embedded in R^3 using persistence diagrams generated from all possible height filtrations (an uncountably infinite number of directions). In this paper, we present an algorithm for reconstructing plane graphs K = (V, E) in R^2, i.e., a planar graph with vertices in general position and a straight-line embedding, from a quadratic number height filtrations and their respective persistence diagrams. 

   more » « less
5. Learning Simplicial Complexes from Persistence Diagrams

   Robin Lynne Belton, Brittany Terese (July 2018, CCCG '18: Thirtieth Canadian Conference on Computational Geometry)

   Topological Data Analysis (TDA) studies the shape of data. A common topological descriptor is the persistence diagram, which encodes topological features in a topological space at different scales. Turner, Mukeherjee, and Boyer showed that one can reconstruct a simplicial complex embedded in R^3 using persistence diagrams generated from all possible height filtrations (an uncountably infinite number of directions). In this paper, we present an algorithm for reconstructing plane graphs K=(V,E) in R^2 , i.e., a planar graph with vertices in general position and a straight-line embedding, from a quadratic number height filtrations and their respective persistence diagrams. 

   more » « less