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1. Pressure tuning of Kitaev spin liquid candidate Na3Co2SbO6

   https://doi.org/10.1038/s42005-025-02174-2  

   Poldi, E_H T; Tartaglia, R; Fabbris, G; Nguyen, N; Park, H; Liu, Z; van_Veenendaal, M; Kumar, R; Jose, G; Samanta, S; et al (December 2025, Communications Physics)

   Abstract The search for Kitaev’s quantum spin liquid in real materials has recently expanded with the prediction that honeycomb lattices of divalent, high-spin cobalt ions could host the dominant bond-dependent exchange interactions required to stabilize the elusive entangled quantum state. The layered honeycomb Na₃Co₂SbO₆has been singled out as a leading candidate provided that the trigonal crystal field acting on Co 3dorbitals, which enhances non-Kitaev exchange interactions between$${J}_{{{\rm{eff}}}}=\frac{1}{2}$$ $J_{\mathrm{eff}}=\frac{1}{2}$spin-orbital pseudospins, is reduced. Here we show that applied pressure leads to anisotropic compression of the layered structure, significantly reducing the trigonal distortion of CoO₆octahedra. Ferromagnetic correlations between pseudospins are enhanced in the spin-polarized (3 Tesla) phase up to about 60 GPa. Higher pressures drive a high-spin to low-spin transition destroying the$${J}_{{{\rm{eff}}}}=\frac{1}{2}$$ $J_{\mathrm{eff}}=\frac{1}{2}$moments required to map the spin Hamiltonian into Kitaev’s model. The spin transition strongly suppresses the low-temperature magnetic susceptibility and appears to stabilize a paramagnetic phase driven by frustration. The possible emergence of frustrated magnetism of localized$$S=\frac{1}{2}$$ $S=\frac{1}{2}$moments opens the door for exploration of novel magnetic quantum states in compressed honeycomb lattices of divalent cobaltates. 

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2. On a Traveling Salesman Problem for Points in the Unit Cube

   https://doi.org/10.1007/s00453-024-01257-w  

   Balogh, József; Clemen, Felix Christian; Dumitrescu, Adrian (September 2024, Algorithmica)

   Abstract LetXbe ann-element point set in thek-dimensional unit cube$$[0,1]^k$$ ${[0,1]}^k$where$$k \ge 2$$ $k\ge 2$. According to an old result of Bollobás and Meir (Oper Res Lett 11:19–21, 1992) , there exists a cycle (tour)$$x_1, x_2, \ldots , x_n$$ $x_1,x_2,\dots ,x_n$through thenpoints, such that$$\left( \sum _{i=1}^n |x_i - x_{i+1}|^k \right) ^{1/k} \le c_k$$ ${\left({\sum }_{i=1}^n,{|x_i-x_{i+1}|}^k\right)}^{1/k}\le c_k$, where$$|x-y|$$ $|x-y|$is the Euclidean distance betweenxandy, and$$c_k$$ $c_k$is an absolute constant that depends only onk, where$$x_{n+1} \equiv x_1$$ $x_{n+1}\equiv x_1$. From the other direction, for every$$k \ge 2$$ $k\ge 2$and$$n \ge 2$$ $n\ge 2$, there existnpoints in$$[0,1]^k$$ ${[0,1]}^k$, such that their shortest tour satisfies$$\left( \sum _{i=1}^n |x_i - x_{i+1}|^k \right) ^{1/k} = 2^{1/k} \cdot \sqrt{k}$$ ${\left({\sum }_{i=1}^n,{|x_i-x_{i+1}|}^k\right)}^{1/k}=2^{1/k}·\sqrt{k}$. For the plane, the best constant is$$c_2=2$$ $c_2=2$and this is the only exact value known. Bollobás and Meir showed that one can take$$c_k = 9 \left( \frac{2}{3} \right) ^{1/k} \cdot \sqrt{k}$$ $c_k=9{\left(\frac{2}{3}\right)}^{1/k}·\sqrt{k}$for every$$k \ge 3$$ $k\ge 3$and conjectured that the best constant is$$c_k = 2^{1/k} \cdot \sqrt{k}$$ $c_k=2^{1/k}·\sqrt{k}$, for every$$k \ge 2$$ $k\ge 2$. Here we significantly improve the upper bound and show that one can take$$c_k = 3 \sqrt{5} \left( \frac{2}{3} \right) ^{1/k} \cdot \sqrt{k}$$ $c_k=3\sqrt{5}{\left(\frac{2}{3}\right)}^{1/k}·\sqrt{k}$or$$c_k = 2.91 \sqrt{k} \ (1+o_k(1))$$ $c_k=2.91\sqrt{k}(1+o_k\left(1\right))$. Our bounds are constructive. We also show that$$c_3 \ge 2^{7/6}$$ $c_3\ge 2^{7/6}$, which disproves the conjecture for$$k=3$$ $k=3$. Connections to matching problems, power assignment problems, related problems, including algorithms, are discussed in this context. A slightly revised version of the Bollobás–Meir conjecture is proposed. 

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3. Hardness and approximation of submodular minimum linear ordering problems

   https://doi.org/10.1007/s10107-023-02038-z  

   Farhadi, Majid; Gupta, Swati; Sun, Shengding; Tetali, Prasad; Wigal, Michael C (December 2023, Mathematical Programming)

   Abstract The minimum linear ordering problem (MLOP) generalizes well-known combinatorial optimization problems such as minimum linear arrangement and minimum sum set cover. MLOP seeks to minimize an aggregated cost$$f(\cdot )$$ $f(·)$due to an ordering$$\sigma $$ $\sigma$of the items (say [n]), i.e.,$$\min _{\sigma } \sum _{i\in [n]} f(E_{i,\sigma })$$ ${min}_{\sigma }{\sum }_{i\in \left[n\right]}f\left(E_{i,\sigma }\right)$, where$$E_{i,\sigma }$$ $E_{i,\sigma }$is the set of items mapped by$$\sigma $$ $\sigma$to indices [i]. Despite an extensive literature on MLOP variants and approximations for these, it was unclear whether the graphic matroid MLOP was NP-hard. We settle this question through non-trivial reductions from mininimum latency vertex cover and minimum sum vertex cover problems. We further propose a new combinatorial algorithm for approximating monotone submodular MLOP, using the theory of principal partitions. This is in contrast to the rounding algorithm by Iwata et al. (in: APPROX, 2012), using Lovász extension of submodular functions. We show a$$(2-\frac{1+\ell _{f}}{1+|E|})$$ $(2-\frac{1+{\ell }_f}{1+\left|E\right|})$-approximation for monotone submodular MLOP where$$\ell _{f}=\frac{f(E)}{\max _{x\in E}f(\{x\})}$$ ${\ell }_f=\frac{f\left(E\right)}{{max}_{x\in E}f\left(\left\{x\right\}\right)}$satisfies$$1 \le \ell _f \le |E|$$ $1\le {\ell }_f\le \left|E\right|$. Our theory provides new approximation bounds for special cases of the problem, in particular a$$(2-\frac{1+r(E)}{1+|E|})$$ $(2-\frac{1+r\left(E\right)}{1+\left|E\right|})$-approximation for the matroid MLOP, where$$f = r$$ $f=r$is the rank function of a matroid. We further show that minimum latency vertex cover is$$\frac{4}{3}$$ $\frac{4}{3}$-approximable, by which we also lower bound the integrality gap of its natural LP relaxation, which might be of independent interest. 

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4. Relating elliptic curve point-counting and solutions of quadratic forms with congruence conditions

   https://doi.org/10.1007/s11139-026-01341-5  

   Mondal, Koustav (March 2026, The Ramanujan Journal)

   Abstract In this paper, we analyze the theta series associated to the quadratic form$$Q(\vec {x}) :=x_1^2+x_2^2+x_3^2+x_4^2$$ $Q\left(\overrightarrow{x}\right):=x_1^2+x_2^2+x_3^2+x_4^2$with congruence conditions on$$x_i$$ $x_i$modulo 2, 3, 4 and 6. By employing special operators on modular, non-holomorphic Eisenstein series of weight 2, we construct a basis for Eisenstein space for levels$$2^k, k\le 7$$ $2^k,k\le 7$,$$3^{\ell }, \ell \le 3$$ $3^{\ell },\ell \le 3$andp, for odd primep. Using the relation between the trace of Frobenius on an elliptic curve and the Fourier coefficients of the cusp form part of theta series corresponding toQ, we establish relation between the number of integer solutions to the equation$$Q(\vec {x}) = p$$ $Q\left(\overrightarrow{x}\right)=p$and the number of$$\mathbb {F}_p$$ $F_p$-rational points on the associated elliptic curve under certain congruence conditions onp. 

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5. Generalized Parking Function Polytopes

   https://doi.org/10.1007/s00026-023-00671-1  

   Hanada, Mitsuki; Lentfer, John; Vindas-Meléndez, Andrés R (June 2024, Annals of Combinatorics)

   Abstract A classical parking function of lengthnis a list of positive integers$$(a_1, a_2, \ldots , a_n)$$ $(a_1,a_2,\dots ,a_n)$whose nondecreasing rearrangement$$b_1 \le b_2 \le \cdots \le b_n$$ $b_1\le b_2\le \cdots \le b_n$satisfies$$b_i \le i$$ $b_i\le i$. The convex hull of all parking functions of lengthnis ann-dimensional polytope in$${\mathbb {R}}^n$$ $R^n$, which we refer to as the classical parking function polytope. Its geometric properties have been explored in Amanbayeva and Wang (Enumer Combin Appl 2(2):Paper No. S2R10, 10, 2022) in response to a question posed by Stanley (Amer Math Mon 127(6):563–571, 2020). We generalize this family of polytopes by studying the geometric properties of the convex hull of$${\textbf{x}}$$ $x$-parking functions for$${\textbf{x}}=(a,b,\dots ,b)$$ $x=(a,b,\cdots ,b)$, which we refer to as$${\textbf{x}}$$ $x$-parking function polytopes. We explore connections between these$${\textbf{x}}$$ $x$-parking function polytopes, the Pitman–Stanley polytope, and the partial permutahedra of Heuer and Striker (SIAM J Discrete Math 36(4):2863–2888, 2022). In particular, we establish a closed-form expression for the volume of$${\textbf{x}}$$ $x$-parking function polytopes. This allows us to answer a conjecture of Behrend et al. (2022) and also obtain a new closed-form expression for the volume of the convex hull of classical parking functions as a corollary. 

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