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arXiv2026

Asymptotic Bounds on Generalized Covering Radii of Binary Primitive BCH Codes

Fix integers $e\ge2$ and $r\ge1$. In this paper we study the $r$-th generalized covering radius $ρ_r\left(BCH(e,m)\right)$ of the binary primitive $e$-error-correcting BCH code $BCH(e,m)$. By using an algebraic-geometric reformulation of the covering problem together with an explicit Lang-Weil estimate, we prove that \[ρ_r\bigl(\BCH(e,m)\bigr)\le(r+1)e-1\] for all sufficiently large $m$. For $e\ge7$, this improves a recent result of Belinsky--Zabokritskiy. Our proof gives a substantially simpler geometric approach to this upper bound. In particular it implies that \[ρ_2\bigl(BCH(e,m)\bigr)=3e-1\] for all sufficiently large $m$. Previously it was only known that \[ρ_2\bigl(\BCH(e,m)\bigr) \in \left\{3e-1,3e\right\}\] for all sufficiently large $m$.

cs.IT↗

arXiv2026

Structural Corrections to the Bethe Approximation of the Permanent

We study deterministic approximation algorithms for the permanent of a nonnegative matrix through the Bethe permanent, an approximation computable in polynomial time. The tight analysis of Anari and Rezaei gives a universal comparison between the permanent and the Bethe permanent within a factor $(\sqrt 2)^n$. The simple example of the unweighted $4$-cycle $C_4$ (or a union of disjoint $C_4$'s) shows that this bound is tight. We show that such $4$-cycle obstructions can be identified and exploited algorithmically. Given a Bethe optimizer, our algorithm identifies nearly isolated weighted $2\times2$ blocks and peels off a vertex-disjoint family of them. If the total weighted correction is large, we can improve the Bethe approximation; if it is small, we show that the Bethe permanent is within a factor of $(\sqrt2 - \varepsilon)^n$ of the truth. Combining these facts, we obtain a deterministic polynomial time $(\sqrt2-\varepsilon)^n$-approximation algorithm for the permanent of an arbitrary nonnegative $n\times n$ matrix, where $\varepsilon>0$ is some absolute constant.

cs.DS↗

arXiv2026

Graphs with Long Pseudosimilarity Chains under Consecutive Vertex Deletions

Pseudosimilar vertices are vertices in distinct automorphism orbits whose deletions produce isomorphic graphs. Classical work has studied the existence, group-theoretic origin, and construction of large sets of such vertices. We ask a different recursive question: how long can one repeatedly delete a vertex that is pseudosimilar at the moment of deletion? We define the pseudosimilarity depth of a graph and construct connected graphs in which this process continues through all but a sublinear number of vertices. A two-clock construction gives a square-root deficit uniformly in the order, while a Chinese-remainder construction with many cyclic clocks yields an infinite family of asymmetric graphs with only a polylogarithmic number of vertices left outside the active chain. The mechanism realizes pseudosimilarity by breaking a long hidden automorphism orbit and enlarging the break one vertex at a time. Thus pseudosimilarity can persist through an asymptotically full sequence of vertex deletions, even though every graph encountered in the main construction is asymmetric.

math.CO↗

arXiv2026

Shannon's problem on the monotonicity of entropy and a Conjecture of Tao

Let $X_1,X_2,\ldots$ be i.i.d. finitely supported random variables in a torsion-free abelian group, and write $S_k=X_1+\cdots+X_k$, and $H(S_k)$ is the Shannon entropy $S_k$, for all $k \ge 1$. We prove that, for every fixed $n\geq1$, \[ H(S_{n+1})-H(S_n) \geq \frac12\log\frac{n+1}{n} -o_{H(X_1)\to\infty}(1), \] uniformly over the ambient group and the input law. This proves a conjecture of Tao [29] in 2010.

math.PR↗

arXiv2026

From b-Coloring to $b^*$-Coloring: Large Girth and Parameterized Complexity

A b-coloring is a proper vertex coloring such that every color class contains a vertex, a so-called b-vertex, which sees all colors in its closed neighborhood. This type of coloring has been intensively studied from both structural and algorithmic point of view. Recently, Zaker [DAM 2025] introduced the notion of a b*-coloring, which is a b-coloring in which there is a vertex that sees a b-vertex of every color in its closed neighborhood. The b*-chromatic number is the maximum integer k such that there is a b*-coloring with k colors. We partially answer a question posed by Zaker and prove that graphs of girth at least 7 are b*-monotonic, which means that the b*-chromatic number does not increase by taking an induced subgraph. In addition, we discover a class of d-regular graphs of girth at least 5 with b*-chromatic number d+1, which strengthens a result about b-colorings by Dettlaff, Furmańczyk, Peterin, Roux, and Ziemann [AMC 2024]. We also study the parameterized complexity of finding b*-colorings, and show that for many structural parameters, the complexity coincides with that of finding b-colorings. In particular, the b*-chromatic number can be computed in polynomial time on any class of bounded clique-width. For most parameters, the translation from b-colorings is straightforward but for the feedback edge number, the FPT algorithm for b*-colorings is actually much simpler than that for b-colorings by Balabán [MFCS 2026].

cs.DM↗

arXiv2026

Ordered Ruzsa-Szemeredi Numbers at Matching Size Two

Bondy and Szwarcfiter defined $\mathrm{ex}^*(n,F)$ as the largest number of edges in an $n$-vertex graph whose edge set partitions into induced copies of $F$; for $F=2K_2$ the deficiency $\binom{n}{2}-\mathrm{ex}^*(n,2K_2)$ is $Θ(n^{3/2})$. We study the ordered relaxation at fixed matching size, in which each part need only be induced in the union of itself with the parts that follow it; write $\mathrm{ORS}_n(r)$ for the largest number of parts, so that $r\,\mathrm{ORS}_n(r)$ is the ordered analogue of $\mathrm{ex}^*(n,rK_2)$. Our main tool is a characterisation valid for every $r$: an ordered decomposition into induced $r$-matchings is a sequence of steps that start from $K_n$ and repeatedly delete a perfect matching from $2r$ vertices currently spanning a clique. Reading a decomposition backwards turns a condition about the ordering into a reachability question that an exhaustive search can settle. For $r=2$ we determine $\mathrm{ORS}_n(2)$ exactly at orders five through nineteen, where it takes the values $1,3,5,8,11,14,19,23,28,34,40,47,54,62,70$, and we confine $\mathrm{ORS}_{20}(2)$ to $\{78,79\}$. The counting bound $\lfloor n(n-4)/4\rfloor$ is attained at orders five through nine and at eleven, and missed by exactly one part at every other order below twenty, so order eleven is an isolated exception, not a parity effect. Across this range the ordered deficiency equals $\frac32n+O(1)$, and along powers of two a dyadic construction keeps it below $O(n\log n)$; whether it is linear for all $n$ is our main open question. The structural results are formalised in Lean 4, and the searches are certified by fail-closed sweeps and an independent checker.

math.CO↗

arXiv2026

FirstFit online coloring in the random order model

The average performance of FirstFit online coloring on trees in the random order model is completely determined in recent works of Frei et al. and Bosek et al., showing $Θ(\log n /\log\log n)$ number of colors, improving the $Θ(\log n)$ colors in the adversarial model. We provide a few further results on slightly more general graph classes. Firstly, we extend their method to obtain a simple path-counting principle for sparse graph classes, which immediately yields for example that cactus graphs and uniform hypertrees exhibit a similar improvement. We then show that FirstFit uses only $O(1)$ colors on crown graphs, a standard example where adversarial arrival forces $Θ(n)$ colors. We further show that density alone (even linear minimum degree) is insufficient to guarantee $O(1)$ colors even on bipartite graphs. Finally, we identify graph classes, including unit interval graphs and some graphs of high chromatic number, for which random arrival provides only limited improvement. We end with some open problems.

cs.DS↗

arXiv2026

Efficient Polynomial-Time Decoding of Simplicial Anticodes with Near-Optimal Performance

In this work, we propose an efficient decoding algorithm for codes arising from simplicial complexes, a family of binary linear codes for which no decoding method of this type was previously known. Although the algorithm does not always attain the maximum theoretical error-correcting capability, it provides an explicit bound that can be computed directly from the structure of the complex. Moreover, this bound is asymptotically optimal: the ratio between the guaranteed correcting capability and the theoretical maximum converges to $1$ as the code length increases, under natural assumptions on the dimension of the maximal faces. The correction capability is also presented in specific examples. Finally, we introduce specific families of simplicial complexes where the algorithm successfully reaches this theoretical bound.

cs.IT↗

arXiv2026

Fair Division of Graphs: Beyond Traceability

In this paper, we study fair division problems in which resources are structured as graphs and agents must receive connected bundles. This connectivity requirement fundamentally alters the problem, making it significantly more challenging than its classical counterpart. We focus on the fairness notion of $\mathrm{EF1}_{\mathrm{outer}}$, where envy can be eliminated by removing at most one vertex whose deletion does not disconnect the bundle -- a critical constraint for applications such as land division and network allocation. Our first result extends prior work by establishing the existence of $\mathrm{EF1}_{\mathrm{outer}}$ allocations for an infinite family of non-traceable graphs (that is, graphs that do not admit a Hamiltonian path), answering a central open question and generalizing Bilò et al.'s result for traceable graphs. We then make progress on a conjecture concerning the $\mathrm{EF1}_{\mathrm{outer}}$ spectrum of trees due to Chen and Zwicker. Finally, we complement our structural results with algorithmic insights, showing that deciding the existence of an $\mathrm{EF1}_{\mathrm{outer}}$ allocation is NP-complete even for binary additive valuations, thereby resolving an open complexity question. Taken together, our results deepen the connection between graph theory and fair division, and offer new tools for studying fairness in structured resource environments.

cs.GT↗

arXiv2026

Mutually orthogonal anti-Latin squares

Anti-Latin squares were introduced in connection with non-linear secure network coding, and the extremal problem for large mutually orthogonal families is motivated by that setting. We study the maximum size $N_A(d)$ of a family of mutually orthogonal anti-Latin squares of order $d$. We prove that $N_L(d)+1\le N_A(d)\le N_L(d)+2$ for every $d\ge 3$, where $N_L(d)$ denotes the classical maximum size of a family of mutually orthogonal Latin squares of order $d$, and we show that in fact $N_A(3)=N_L(3)+1$ whereas $N_A(d)=N_L(d)+2$ for every $d\ge 4$. The upper bound is obtained by passing through balanced matrices, while the lower bound is given by a deterministic permutation argument. For all $d\ge 8$, and also for the exceptional order $d=6$, the upper bound is shown to be attainable by a general probabilistic construction. On the structural side, we show that a saturated family of size $d+1$ induces an affine plane of order $d$, and that the saturated case is characterized by the existence of an anti-coordinate grid decomposition; after transporting this condition to the fixed cell set $[d]^2$, it becomes a direction-completeness condition on the corresponding row-blocks and column-blocks. The remaining small orders are treated separately: $d=3$ is handled by direct analysis and classification of orthogonal triples, $d=4$ by an explicit saturated construction and an analysis of its finite-geometric structure, and $d=5$ and $d=7$ by explicit saturated examples arising from the random-grid framework. Thus $N_A(d)$ is determined in terms of $N_L(d)$ for every $d\ge3$, and its numerical value is obtained explicitly for every $3\le d\le9$.

math.CO↗

arXiv2026

Distinguishing classes of intersection graphs of homothets or similarities of two convex disks

For smooth convex disks $A$, i.e., convex compact subsets of the plane with non-empty interior and with at most one tangent at every boundary point, we classify the classes $G^{\text{hom}}(A)$ and $G^{\text{sim}}(A)$ of intersection graphs that can be obtained from homothets and similarities of $A$, respectively. Namely, we prove that $G^{\text{hom}}(A)=G^{\text{hom}}(B)$ if and only if $A$ and $B$ are affine equivalent, and $G^{\text{sim}}(A)=G^{\text{sim}}(B)$ if and only if $A$ and $B$ are similar.

cs.CG↗

arXiv2026

Constructive Characterization and Recognition Algorithm for Grafts with a Connected Minimum Join

Minimum joins in a graft $(G, T)$, also known as minimum $T$-joins of a graph $G$, are said to be connected if they determine a connected subgraph of $G$. Grafts with a connected minimum join have gained interest ever since Middendorf and Pfeiffer showed that they satisfy Seymour's min-max formula for joins and $T$-cut packings; that is, in such grafts, the size of a minimum join is equal to the size of a maximum packing of $T$-cuts. In this paper, we provide a constructive characterization of grafts with a connected minimum join. We also obtain a polynomial time algorithm that decides whether a given graft has a connected minimum join and, if so, outputs one. Our algorithm has two bottlenecks; one is the time required to compute a minimum join of a graft, and the other is the time required to solve the single-source all-sink shortest path problem in a graph with conservative $\pm 1$-valued edge weights. Thus, our algorithm runs in $O(n(m + n\log n) )$ time. In the nondense case, it improves upon the time bound for this problem due to Sebő and Tannier that was introduced as an application of their results on metrics on graphs.

cs.DM↗

arXiv2026

Column Number of Delta-modular matrices: Refined Analysis via Sauer Matrices

In this paper, we build upon the analysis initiated by Gennadiy Averkov and Matthias Schymura (2022) and establish that the number of distinct columns of a $Δ$-modular matrix $A \in \mathbb{Z}^{m \times n}$ of rank $m$ is $O(m^3 Δ)$. This upper bound was previously known only for odd values of $Δ$. Recall that a matrix is called $Δ$-modular if the maximum of the absolute values of its $m \times m$ minors equals $Δ$.

math.CO↗

arXiv2026

It's Hard to PArcK

We show that Partizan Arc Kayles (PArcK), a generalization of Domineering to graphs, is PSPACE-complete via a reduction from Positive CNF and with recently-discovered techniques for creating PArcK positions with high temperature. The reduction uses only red and blue edges.

cs.CC↗

arXiv2026

Further Comments on Yablo's Construction

We continue our analysis of Yablo's coding of the liar paradox by infinite acyclic graphs. The present notes are based on and continue the author's previous results on the problem. In particular, our approach is often more systematic than before.

math.CO↗

arXiv2026

Double Toeplitz codes and their average weight enumerators

Recently, double Toeplitz codes have been introduced as a generalization of double circulant codes. In this paper, we study the average weight enumerators of double Toeplitz codes. As an application, we consider the existence of double Toeplitz codes over $\mathbb{F}_q$ with some specified minimum weights for $q \in \{2,3,4\}$. We also give a classification of double Toeplitz codes over $\mathbb{F}_q$ with the largest minimum weights for moderate lengths and $q \in \{2,3,4\}$.

math.CO↗

arXiv2026

Entropy lower bounds and sum-product phenomena

Various lower bounds are established for the entropy of sums, products and their combinations. First, we derive a prime-field analogue of a version of the entropy power inequality established by Tao over torsion-free groups. Next, we prove an entropy sum-product statement: For independent and identically distributed random variables $X,X'$, the maximum of ${\bf H}(X+X')$ and ${\bf H}(XX')$ is bounded below by a linear combination of the entropy and the min-entropy (Rényi entropy of order~$\infty$) of $X$. This result, obtained by bounding entropies of the form ${\bf H}\bigl( X(Y+Z)\bigr)$ from above and below, is valid over arbitrary fields $F$. Over $F={\bf R}$, a slightly stronger inequality is derived. Finally, a weak version of a purely Shannon-entropic sum-product result is developed: If the entropic additive doubling of a random variable $X$ over an arbitrary field is $O(1)$, then its multiplicative doubling is at least proportional to ${\bf H}(X)$.

math.CO↗

arXiv2026

Fine Difference Structure and Prime-Power Depth of Bent Partitions

A $p$-ary bent partition of $\mathbb{F}_p^n$ is a partition into $K$ nonempty cells such that every balanced assignment of its cells to $\mathbb{F}_p$ produces a bent function. It was asked whether every possible depth $K$ is a power of $p$; for general $p$, previous affirmative results required regularity or cell-symmetry hypotheses. We prove the stronger unconditional statement that, for every nonzero $h$, exactly $p^n/K$ points remain in the same fine cell under translation by $h$. Thus the fine cells form a partitioned difference family and the fine label map is zero-difference balanced. Consequently $K\mid p^n$, so $K=p^t$; nonempty cells further give $1\le t<n$. In even dimension, the classical cell-size theorem yields $K\mid p^{n/2}$. Together with the known odd-dimensional ternary three-fibre parameter restriction, this gives the global bound $t\le\lfloor n/2\rfloor$. The proof is an exact finite average over balanced coarsenings. The main counting identity and selected consequences are formalized and kernel-checked in Lean 4.

cs.IT↗