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arXiv2026

The fourth generalized Davenport constant of $C_5^3$

For a finite abelian group $G$ and $k \geq 1$, the generalized Davenport constant $D_k(G)$ is the least $\ell$ such that every sequence over $G$ of length at least $\ell$ has $k$ pairwise disjoint nonempty zero-sum subsequences. A theorem of Freeze and Schmid gives $D_k(C_5^3) \geq 5k+10$ for every $k \geq 2$. We prove the matching upper bound: $D_4(C_5^3)=30$, and hence $D_k(C_5^3)=5k+10$ for every $k \geq 2$, so the Freeze--Schmid bound is attained by $C_5^3$ from $k=2$ onward, as it is by $C_2^3$ and unlike $C_3^3$. The proof is finite and computer-assisted. The remaining case reduces to showing that every zero-sum sequence of length $31$ over $C_5^3$ contains a nonempty zero-sum subsequence of length at most five. A saturation argument confines the multiplicities of a hypothetical counterexample to $\{1,2,4\}$, its support pattern to one of $60$ solutions of two linear equations, and its geometry to one of $78$ rank/plane branches normalized to a standard basis; an exhaustive search exhausts every branch with no survivor. The search was carried out by three independently written implementations, and the branch cover was regenerated by separate programs from the lemmas alone; two further machine-verified values, $D_3(C_5^3)=25$ and $s_{\leq 6}(C_5^3)=24$, enter the second statement, and their records accompany the paper.

cs.DM↗

arXiv2026

The Erdős-Pósa Property for Colorful Minors

A colorful graph relation enhances the minor relation by merging color sets along contractions and by allowing the removal of colors; it generalizes rooted minors and models problems on graphs with several, possibly overlapping, annotated vertex sets. A graph has the Erdős-Pósa property for minors if and only if it is planar, by a classical theorem of Robertson and Seymour. In this work we determine, for the colorful minor relation, exactly which colorful graphs have the Erdős-Pósa property. Our characterization takes three equivalent forms. The first is structural: the colorful graphs with the property are those that can be drawn with all their colored vertices on one face and whose colors are, in a precise sense, laid out along that face without interleaving. The second is given by an obstruction set: they are those excluding every member of an explicit infinite family $\mathcal{O},$ of which only $\mathbf{O}(|I|^{4})$ members have colors that are a subset of $I,$ for every finite set $I$ of colors. The third is grid-like: they are exactly the colorful minors of unions of particular families of segregated grids, the colorful analogues of the grids that drive the classical proof.

math.CO↗

arXiv2026

Order 14 is the largest order for which every 4-total coloring of every cubic graph is equitable

A total coloring of a graph is an assignment of colors to its vertices and edges so that adjacent or incident elements receive distinct colors, and it is equitable when the cardinalities of any two color classes differ by at most one. Stemock conjectured that every $4$-total coloring of a cubic graph of order less than $20$ is equitable. In this paper, we disprove this conjecture: the circular ladder $L_{12}$ admits a non-equitable $4$-total coloring and, moreover, no smaller counterexample exists: order $4$ is vacuous, and every $4$-total coloring of a cubic graph of order $6$, $8$, or $10$ is equitable. We also prove that the same property holds at order $14$. Our proofs rely on a decomposition lemma, which states that, in any $4$-total coloring of a cubic graph $G$, each color class consists of an independent set $S$ together with a perfect matching of $G-S$. We use the lemma to determine all possible color class configurations for orders $12$, $16$, and $18$, and we show that every listed configuration is attained. Finally, we provide a splicing construction showing that, for every even $n\geq16$, some connected cubic graph of order $n$ admits a non-equitable $4$-total coloring. We may conclude that $14$ is the largest order for which every $4$-total coloring of every cubic graph is equitable.

math.CO↗

arXiv2026

Integrality Gap Bounds for the Goemans-Linial SDP on Finite Abelian Cayley Graphs

In the uniform sparsest cut problem we are asked to find a vertex set that cuts few edges relative to the number of vertex pairs it separates. The Goemans-Linial SDP coupled with the Arora-Rao-Vazirani rounding gives an $\mathcal{O}(\sqrt{\log n})$ approximation on arbitrary graphs on $n$ vertices. We study this relaxation on finite Abelian Cayley graphs. First we show that when the second normalized Laplacian eigenvalue of $G= \mathrm{Cayley}(Γ, S)$ is realized by a Fourier character with image size at most four then $λ_2(G)=\mathrm{SDP}_{\mathrm{GL}}(G)=ψ(G)$. Geometrically, a character maps the vertices onto a regular polygon where the squared chord distance satisfies the triangle inequalities exactly when the polygon has at most four vertices. Grouping equal character fibers gives a cyclic quotient where the optimal cut can be found exactly and so the relaxation is exact on finite Abelian Cayley graphs on groups of exponent at most four. Second, we replace each generator $s$ of $S$ by a uniformly random element of its cyclic subgroup (including identity). If $r_s$ is the order of $s$, we let $α(r_s)$ to be the average number of $\pm s$ steps needed to simulate such a move, and let $ρ(S)=\max_{s\in S}α(r_s)$ be its worst case. Full cyclic averaging eliminates character phases and choosing a nontrivial character $χ^*$ minimizing the auxiliary eigenvalue and taking $K=\mathrm{ker}χ^*$ gives \[ ψ(G)\leqψ_G(K)\leq\frac{q^*}{q^*-1} \cdotρ(S)\cdot\mathrm{SDP}_{\mathrm{GL}}(G)\leq 2ρ(S)\cdot\mathrm{SDP}_{\mathrm{GL}}(G), \] where $q^*=|χ^*(Γ)|$. If all generator orders are at most $R$, this is an $R/2$ approximation. Finally, we construct an infinite family of finite Abelian Cayley graphs with Goemans-Linial integrality gap exactly $16/15$.

math.CO↗

arXiv2026

Optimal mixing of the systematic scan dynamics via approximate tensorization of entropy

We study the mixing time of the systematic scan dynamics for high-dimensional discrete distributions. This Markov chain updates coordinates sequentially according to a fixed predetermined order, in contrast to the Glauber dynamics that updates coordinates selected uniformly at random. The systematic scan is often favored in practice because it exhibits strong empirical performance, but its theoretical analysis remains far less developed than that of Glauber dynamics. We take a step toward addressing this imbalance by showing that two standard functional notions of weak dependence between the coordinates of the distribution provide strong convergence guarantees for the systematic scan dynamics. First, we show that approximate tensorization of entropy implies optimal $O(\log n)$ mixing time for every scan order under standard marginal, connectivity, and bounded interaction degree assumptions about the distribution. Second, we show that approximate tensorization of variance yields a constant-factor contraction of the variance functional per scan, which in turn implies an optimal $O(1)$ relaxation time for the natural additive and multiplicative reversibilizations of the systematic scan dynamics. Compared with our entropy result, the variance bound improves the dependence on the maximum interaction degree from exponential to quadratic and requires weaker assumptions on the distribution. As concrete applications of our results, we establish optimal $O(\log n)$ mixing of the systematic scan dynamics for bounded-degree antiferromagnetic two-spin systems in the tree-uniqueness region and for the ferromagnetic $q$-state Potts model on square boxes in $\mathbb{Z}^2$ throughout its subcritical regime.

math.PR↗

arXiv2026

Moments of crosscorrelation demerit factors of binary sequences

Families of sequences with low mutual aperiodic crosscorrelation assist the design of systems for multi-user asynchronous communications and multiple-input multiple-output radar. The crosscorrelation demerit factor of a pair of sequences is the sum of the squared magnitudes of their crosscorrelation values at every shift when the sequences are normalized to unit Euclidean norm, and the merit factor is the reciprocal of the demerit factor. For each positive integer $\ell$, we endow the $2^{2 \ell}$ pairs of binary sequences of length $\ell$ with uniform probability measure and study the distribution of their crosscorrelation demerit factors. Sarwate showed that the mean value is always $1$ regardless of length $\ell$. We develop a method for finding an exact formula for the $p$th central moment (for any positive integer $p$) as a function of $\ell$. Formulae for the variance and third central moment ($p=2$ and $3$) are then obtained by hand calculations, while the fourth through sixth central moments are obtained by computer-assisted calculations. Our theory also shows that all the central moments must be strictly positive for $p\geq 2$ and $\ell \geq 3$.

cs.IT↗

arXiv2026

Equality cases of the Stanley--Yan log-concave matroid inequality

The \emph{Stanley--Yan} (SY) \emph{inequality} gives the ultra-log-concavity for the numbers of bases of a matroid which have given sizes of intersections with $k$ fixed disjoint sets. The inequality was proved by Stanley (1981) for regular matroids, and by Yan (2023) in full generality. In the original paper, Stanley asked for equality conditions of the SY~inequality, and proved total equality conditions for regular matroids in the case $k=0$. In this paper, we completely resolve Stanley's problem. First, we obtain an explicit description of the equality cases of the SY inequality for $k=0$, extending Stanley's results to general matroids and removing the ``total equality'' assumption. Second, for $k\ge 1$, we prove that the equality cases of the SY inequality cannot be described in a sense that they are not in the polynomial hierarchy unless the polynomial hierarchy collapses to a finite level.

math.CO↗

arXiv2026

Long induced paths in sparse graphs and graphs with forbidden patterns

Consider a graph $G$ with a path $P$ of order $n$. What conditions force $G$ to also have a long induced path? As complete bipartite graphs have long paths but no long induced paths, a natural restriction is to forbid some fixed complete bipartite graph $K_{t,t}$ as a subgraph. In this case we show that $G$ has an induced path of order $(\log \log n)^{1/5-o(1)}$. This is an exponential improvement over a result of Galvin, Rival, and Sands (1982) and comes close to a recent upper bound of order $O((\log \log n)^2)$. Another way to approach this problem is by viewing $G$ as an ordered graph (where the vertices are ordered according to their position on the path $P$). From this point of view it is most natural to consider which ordered subgraphs need to be forbidden in order to force the existence of a long induced path. Focusing on the exclusion of ordered matchings, we improve or recover a number of existing results with much simpler proofs, in a unified way. We also show that if some forbidden ordered subgraph forces the existence of a long induced path in $G$, then this induced path has size at least $Ω((\log \log \log n)^{1/3})$, and can be chosen to be increasing with respect to $P$.

math.CO↗

arXiv2026

Parameterized Hardness of Zonotope Containment and Neural Network Verification

Neural networks with ReLU activations are a widely used model in machine learning. It is thus important to have a profound understanding of the properties of the functions computed by such networks. Recently, there has been increasing interest in the (parameterized) computational complexity of determining these properties. In this work, we close several gaps and resolve an open problem posed by Froese et al. [COLT '25] regarding the parameterized complexity of various problems related to network verification. In particular, we prove that, for all $\ell\ge 2$, deciding positivity (and thus surjectivity) of a function $f:\mathbb{R}^d\to\mathbb{R}$ computed by an $\ell$-layer ReLU network is W[$\ell-1$]-hard when parameterized by the input dimension $d$. The case $\ell=2$ implies that zonotope non-containment (a problem that is of independent interest in computational geometry, control theory, and robotics) is W[1]-hard with respect to the ambient dimension $d$. Moreover, we show that approximating the maximum within any multiplicative factor and computing the $L_p$-Lipschitz constant for $p\in(0,\infty]$ in $\ell$-layer networks is NP-hard and W[$\ell-1$]-hard with respect to $d$. For $\ell\ge 3$, approximating the $L_p$-Lipschitz constant is NP- and W[$\ell-2$]-hard. We further show that the above problems are NP- and W[$t$]-hard (for all $t\ge 1$) with respect to $\ell$ for constant $d$. Notably, our hardness results imply that the naive enumeration-based methods for these fundamental problems running in $n^{(\ell-1) d}\cdot\operatorname{poly}(N)$ time are all essentially optimal under the Exponential Time Hypothesis.

cs.CC↗

arXiv2026

A proof of Ross's conjecture for two-site moving-target search

A target moves between two sites according to a discrete-time Markov chain with a $2\times2$ transition matrix $M$. At each epoch one site is searched at positive cost, and a search may overlook a target that is present. Ross conjectured that an optimal policy is threshold in the posterior probability that the target is at site~1. MacPhee and Jordan proved the conjecture throughout the nonpositive-determinant ($\det M\le0$) regime and for part of the positive-determinant ($\det M>0$) regime, leaving the remaining cases open. We prove threshold optimality throughout the positive-determinant regime, completing Ross's conjecture for all parameter values.

math.PR↗

arXiv2026

On the Laplacian spectral gap of generalized pancake graphs

The generalized pancake graph $P(m,n)$ is the Cayley graph of the group of colored permutations $\mathbb{Z}_m\wr S_n=(\mathbb{Z}_m)^n\rtimes S_n$ generated by generalized prefix reversals. In this paper, we establish that, for all $m,n\geq2$, the spectral gap $γ(P(m,n))$ of the normalized Laplacian satisfies $α_m/n\leqγ(P(m,n))\leq1/n$, where $α_m$ is a positive constant that depends only on $m$. As a consequence, for every fixed $m\geq2$, $γ(P(m,n))$ is $Θ_m(1/n)$ as $n\to\infty$. The proof combines Cesi's semi-recursive spectral-gap inequality with a Fourier decomposition of the appropriate operators associated with a coset Schreier graph of color-position pairs. For fixed $n\geq2$, we also establish that $γ(P(m,n))$ is $Θ_n(m^{-2})$ as $m\to\infty$. This disproves a conjecture of Blanco and Buehrle asserting that, for fixed $n$, the corresponding undirected generalized pancake graphs form an expander family. Additionally, we present a counterexample to a recent conjecture of Greaves and Zhu concerning equality between the spectral gaps of the full Cayley graph and the associated coset Schreier graph.

math.CO↗

arXiv2026

Helly-Type Theorems for Splitting Point Sets

Let $0 < α\leq 1/2$. We say that a finite point set $P$ in $\mathbb{R}^d$ is $α$-split by a hyperplane $h$ if each of the closed half-spaces determined by $h$, contains at least $α|P|$ of the points of $P$. We further say $P$ is $α$-split by a $k$-dimensional flat $τ$ if $P$ is $α$-split by any hyperplane through $τ$. In the standard notation (which coincides with Tukey depth for $k= 0$), the $k$-flat $τ$ has depth $α$ with respect to $P$. We establish interesting Helly-type theorems for splitting families of finite point sets in $\mathbb{R}^d$. Unlike the classical sufficient Helly-type criteria for transversals to families of compact convex sets, which exist only for point and hyperplanes, our results extend to splitting families of point sets by collections of $k$-flats of arbitrary dimensionality $ 0 \leq k \leq d-1$.

math.CO↗

arXiv2026

Nearly Tight Bounds for Proportional Group Fair Divisions and One-Sided Discrepancy

This paper studies the problem of fair division of indivisible goods among $k$ groups of $n_1,\ldots, n_k$ agents. We look at the worst downward deviation $\textit{PROP}(n_1,\ldots, n_k)$ of an agent in a group from its $1/k$-share. We improve the bounds of (Manurangsi and Meka, 2026) and show that $\textit{PROP}(n_1,\ldots, n_k) = \tildeΘ(\sqrt{n/k})$, where $n = n_1 + \ldots + n_k$ is the total number of agents. For the proof of the upper bound, we develop novel discrepancy-type tools and, in particular, a way to efficiently work with one-sided discrepancy constraints.

cs.DM↗

arXiv2026

The Prime Clockwork: A Dynamic Representation of Modular and Multiplicative Arithmetic

The way numbers are represented strongly influences which arithmetic structures are easy to see. The \emph{prime clockwork} is a recursively growing discrete dynamical system: a list of autonomous two-hand clocks driven by one common $+1$ signal. No primes or primality labels are supplied. Starting empty, the process appends a clock of period $n$ whenever none already present rings; the primes are generated internally as its growth times. For each installed prime $p$, the seconds reading $R_p$ advances through $0,\ldots,p-1$, and each return to zero increments the minutes reading $M_p$, which counts completed $p$-cycles. The hands use only increment, comparison, reset, and carry, without explicit \texttt{mod} or \texttt{div} operations. At time $n$, $n=pM_p(n)+R_p(n)$. The valuation readout $V_p(n)=ν_p(n)$ is generated locally: it is zero when the seconds counter is non-zero (silent state) and otherwise (when the p-clock rings) one plus the earlier valuation addressed by the current minutes reading. The valuation vector gives the integer in unique prime-factorized form. Its coordinates add and subtract under multiplication and division, representing every positive rational uniquely; divisibility becomes weak componentwise order, and unique factorization is natural in this representation. Finite seconds arrays form Cartesian-product state spaces whose common orbit visits every joint state once before repeating; this \emph{grand cycle} is the order-sensitive dynamical counterpart of the Chinese remainder theorem. The same coordinates expose gcd, lcm, perfect powers, Bézout's identity, and Euler's totient. Rational valuation levels reach certain positive algebraic irrationalities, but not algebraic numbers in general.

math.HO↗

arXiv2026

Curves, points, incidences and covering

Given a point set, mostly a grid in our case, we seek upper and lower bounds on the number of curves that are needed to cover the point set. We say a curve covers a point if the curve passes through the point. We consider such coverings by monotonic curves, lines, orthoconvex curves, circles, etc. We also study a problem that is converse of the covering problem -- if a set of $n^2$ points in the plane is covered by $n$ lines then can we say something about the configuration of the points?

math.CO↗

arXiv2026

Tight bounds on the number of non-equivalent parameterized squares in a word

Two words $x,y$ of the same length are said to be \emph{parameterized equivalent} if there exists a character bijection that transforms $x$ into $y$. A word $w$ is called a parameterized square if $w$ is a concatenation of two parameterized equivalent words. Kociumaka et al. [TCS 2016] showed that in a word of length $n$ that contains $σ$ distinct characters, the number of \emph{parameterized squares} that are non-equivalent with respect to parameterized equivalence is at most $2 σ! n$. In this paper, we show that the maximum number of non-equivalent parameterized squares is less than $σn$, which significantly improves the best-known upper bound by Kociumaka et al. Moreover, we construct a family of words containing $Ω(σn)$ non-equivalent parameterized squares, which demonstrates that the upper bound is asymptotically tight.

cs.DS↗

arXiv2026

Multiset Metric Dimension of Binomial Random Graphs

For a graph $G = (V,E)$ and a subset $R \subseteq V$, we say that $R$ is \textit{multiset resolving} for $G$ if for every pair of vertices $v,w$, the \textit{multisets} $\{d(v,r): r \in R\}$ and $\{d(w,r):r \in R\}$ are distinct, where $d(x,y)$ is the graph distance between vertices $x$ and $y$. The \textit{multiset metric dimension} of $G$ is the size of a smallest set $R \subseteq V$ that is multiset resolving (or $\infty$ if no such set exists). This graph parameter was introduced by Simanjuntak, Siagian, and Vitrík in 2017~\cite{simanjuntak2017multiset}, and has since been studied for a variety of graph families. We prove bounds which hold with high probability for the multiset metric dimension of the binomial random graph $G(n,p)$ in the regime $d = (n-1)p = Θ(n^{x})$ for fixed $x \in (0,1)$.

math.CO↗

arXiv2026

Circular Chromatic Numbers, Signability, Relation Algebras, and Network Satisfaction Problems

In this paper, we characterize finite graphs with circular chromatic number less than 3 in terms of the existence of certain signings ($\mathbb Z_2$-labellings studied in the context of signed graphs). In fact, we construct a signed graph which is universal for all such signings -- called anti-triangle-signings in this paper -- of finite $\overline{K_3}$-free graphs, and is closely related to the generic circular triangle-free graph studied by Bodirsky and Guzmán-Pro. Moreover, our universal structure gives rise to a representation of the relation algebra $56_{65}$. We then use this representation to show that the network satisfaction problem described by this relation algebra belongs to NP. This concludes the full classification of the existence of a universal square representation, as well as the complexity of the corresponding network satisfaction problem, for relation algebras with at most four atoms.

math.CO↗