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1. Tight Bounds on the Number of Closest Pairs in Vertical Slabs

   https://doi.org/10.4230/LIPIcs.WADS.2025.8  

   Biniaz, Ahmad; Bose, Prosenjit; Chung, Chaeyoon; De_Carufel, Jean-Lou; Iacono, John; Maheshwari, Anil; Odak, Saeed; Smid, Michiel; Tóth, Csaba D (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Morin, Pat; Oh, Eunjin (Ed.)

   Let S be a set of n points in ℝ^d, where d ≥ 2 is a constant, and let H₁,H₂,…,H_{m+1} be a sequence of vertical hyperplanes that are sorted by their first coordinates, such that exactly n/m points of S are between any two successive hyperplanes. Let |A(S,m)| be the number of different closest pairs in the {(m+1) choose 2} vertical slabs that are bounded by H_i and H_j, over all 1 ≤ i < j ≤ m+1. We prove tight bounds for the largest possible value of |A(S,m)|, over all point sets of size n, and for all values of 1 ≤ m ≤ n. As a result of these bounds, we obtain, for any constant ε > 0, a data structure of size O(n), such that for any vertical query slab Q, the closest pair in the set Q ∩ S can be reported in O(n^{1/2+ε}) time. Prior to this work, no linear space data structure with sublinear query time was known. 

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2. Constraining the clustering transition for colorings of sparse random graphs

   Anastos, M.; Frieze, A.; Pegden, W. (January 2018, The electronic journal of combinatorics)

   Let $$\Omega_q$$ denote the set of proper $[q]$-colorings of the random graph $$G_{n,m}, m=dn/2$$ and let $$H_q$$ be the graph with vertex set $$\Omega_q$$ and an edge $$\set{\s,\t}$$ where $$\s,\t$$ are mappings $$[n]\to[q]$$ iff $$h(\s,\t)=1$$. Here $$h(\s,\t)$$ is the Hamming distance $$|\set{v\in [n]:\s(v)\neq\t(v)}|$$. We show that w.h.p. $$H_q$$ contains a single giant component containing almost all colorings in $$\Omega_q$$ if $$d$$ is sufficiently large and $$q\geq \frac{cd}{\log d}$$ for a constant $c>3/2$. 

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3. Failed zero forcing and critical sets on directed graphs

   Adams, Alyssa; Jacob, Bonnie (October 2021, The Australasian journal of combinatorics)

   Let $$D$$ be a simple digraph (directed graph) with vertex set $V(D)$ and arc set $A(D)$ where $n=|V(D)|$, and each arc is an ordered pair of distinct vertices. If $$(v,u) \in A(D)$$, then $$u$$ is considered an \emph{out-neighbor} of $$v$$ in $$D$$. Initially, we designate each vertex to be either filled or empty. Then, the following color change rule (CCR) is applied: if a filled vertex $$v$$ has exactly one empty out-neighbor $$u$$, then $$u$$ will be filled. If all vertices in $V(D)$$ are eventually filled under repeated applications of the CCR, then the initial set is called a \emph{zero forcing set} (ZFS); if not, it is a \emph{failed zero forcing set} (FZFS). We introduce the \emph{failed zero forcing number} $$\F(D)$$ on a digraph, which is the maximum cardinality of any FZFS. The \emph{zero forcing number}, $$\Z(D)$, is the minimum cardinality of any ZFS. We characterize digraphs that have $$\F(D)<\Z(D)$$ and determine $$\F(D)$$ for several classes of digraphs including de Bruijn and Kautz digraphs. We also characterize digraphs with $$\F(D)=n-1$$, $$\F(D)=n-2$$, and $$\F(D)=0$$, which leads to a characterization of digraphs in which any vertex is a ZFS. Finally, we show that for any integer $$n \geq 3$$ and any non-negative integer $$k$$ with $k 

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4. Semi-Algebraic Off-Line Range Searching and Biclique Partitions in the Plane

   https://doi.org/10.4230/LIPICS.SOCG.2024.4  

   Agarwal, Pankaj K; Ezra, Esther; Sharir, Micha (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Mulzer, Wolfgang; Phillips, Jeff M (Ed.)

   Let P be a set of m points in ℝ², let Σ be a set of n semi-algebraic sets of constant complexity in ℝ², let (S,+) be a semigroup, and let w: P → S be a weight function on the points of P. We describe a randomized algorithm for computing w(P∩σ) for every σ ∈ Σ in overall expected time O^*(m^{2s/(5s-4)}n^{(5s-6)/(5s-4)} + m^{2/3}n^{2/3} + m + n), where s > 0 is a constant that bounds the maximum complexity of the regions of Σ, and where the O^*(⋅) notation hides subpolynomial factors. For s ≥ 3, surprisingly, this bound is smaller than the best-known bound for answering m such queries in an on-line manner. The latter takes O^*(m^{s/(2s-1)}n^{(2s-2)/(2s-1)} + m + n) time. Let Φ: Σ × P → {0,1} be the Boolean predicate (of constant complexity) such that Φ(σ,p) = 1 if p ∈ σ and 0 otherwise, and let Σ_Φ P = {(σ,p) ∈ Σ× P ∣ Φ(σ,p) = 1}. Our algorithm actually computes a partition ℬ_Φ of Σ_Φ P into bipartite cliques (bicliques) of size (i.e., sum of the sizes of the vertex sets of its bicliques) O^*(m^{2s/(5s-4)}n^{(5s-6)/(5s-4)} + m^{2/3}n^{2/3} + m + n). It is straightforward to compute w(P∩σ) for all σ ∈ Σ from ℬ_Φ. Similarly, if η: Σ → S is a weight function on the regions of Σ, ∑_{σ ∈ Σ: p ∈ σ} η(σ), for every point p ∈ P, can be computed from ℬ_Φ in a straightforward manner. We also mention a few other applications of computing ℬ_Φ. 

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5. Convex polytopes in restricted point sets in R^d

   https://doi.org/10.19086/aic.2025.1  

   Bukh, Boris; Zichao, Dong (January 2025, Advances in Combinatorics)

   For a finite point set P⊂R^d, denote by diam(P) the ratio of the largest to the smallest distances between pairs of points in P. Let c_{d,α}(n) be the largest integer c such that any n-point set P⊂R^d in general position, satisfying diam(P)<αn^{1/d}, contains an c-point convex independent subset. We determine the asymptotics of c_{d,α}(n) as n→∞ by showing the existence of positive constants β=β(d,α) and γ=γ(d) such that βn^{(d−1)/(d+1)}≤c_{d,α}(n)≤γn^{(d−1)/(d+1)} for α≥2. 

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