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1. 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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2. On Single-Distance Graphs on the Rational Points in Euclidean Spaces

   https://doi.org/10.4153/S0008439520000181  

   Bau, Sheng; Johnson, Peter; Noble, Matt (March 2021, Canadian Mathematical Bulletin)

   null (Ed.)

   Abstract For positive integers n and d > 0, let $$G(\mathbb {Q}^n,\; d)$$ denote the graph whose vertices are the set of rational points $$\mathbb {Q}^n$$ , with $$u,v \in \mathbb {Q}^n$$ being adjacent if and only if the Euclidean distance between u and v is equal to d . Such a graph is deemed “non-trivial” if d is actually realized as a distance between points of $$\mathbb {Q}^n$$ . In this paper, we show that a space $$\mathbb {Q}^n$$ has the property that all pairs of non-trivial distance graphs $$G(\mathbb {Q}^n,\; d_1)$$ and $$G(\mathbb {Q}^n,\; d_2)$$ are isomorphic if and only if n is equal to 1, 2, or a multiple of 4. Along the way, we make a number of observations concerning the clique number of $$G(\mathbb {Q}^n,\; d)$$ . 

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3. 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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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. Non-covalent assembly of proton donors and p- benzoquinone anions for co-electrocatalytic reduction of dioxygen

   https://doi.org/10.1039/D1SC01271A  

   Hooe, Shelby L.; Cook, Emma N.; Reid, Amelia G.; Machan, Charles W. (July 2021, Chemical Science)

   The two-electron and two-proton p -hydroquinone/ p -benzoquinone (H 2 Q/BQ) redox couple has mechanistic parallels to the function of ubiquinone in the electron transport chain. This proton-dependent redox behavior has shown applicability in catalytic aerobic oxidation reactions, redox flow batteries, and co-electrocatalytic oxygen reduction. Under nominally aprotic conditions in non-aqueous solvents, BQ can be reduced by up to two electrons in separate electrochemically reversible reactions. With weak acids (AH) at high concentrations, potential inversion can occur due to favorable hydrogen-bonding interactions with the intermediate monoanion [BQ(AH) m ]˙ − . The solvation shell created by these interactions can mediate a second one-electron reduction coupled to proton transfer at more positive potentials ([BQ(AH) m ]˙ − + n AH + e − ⇌ [HQ(AH) (m+n)−1 (A)] 2− ), resulting in an overall two electron reduction at a single potential at intermediate acid concentrations. Here we show that hydrogen-bonded adducts of reduced quinones and the proton donor 2,2,2-trifluoroethanol (TFEOH) can mediate the transfer of electrons to a Mn-based complex during the electrocatalytic reduction of dioxygen (O 2 ). The Mn electrocatalyst is selective for H 2 O 2 with only TFEOH and O 2 present, however, with BQ present under sufficient concentrations of TFEOH, an electrogenerated [H 2 Q(AH) 3 (A) 2 ] 2− adduct (where AH = TFEOH) alters product selectivity to 96(±0.5)% H 2 O in a co-electrocatalytic fashion. These results suggest that hydrogen-bonded quinone anions can function in an analogous co-electrocatalytic manner to H 2 Q. 

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