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math.CO: explore 146 source-linked works published from 2024 to 2026, with original documents and citations.

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Includes records with this source-supplied label or an explicit phrase match in their metadata. Matches indicate a mention, not proof that a paper uses a method or tests a material. Source versions are consolidated by DOI.

Sources: arxiv. Collection updated 2026-09-16. Counts describe this index, not the complete source archives.

arXiv2026

An analogue of Reed's conjecture for digraphs

Reed in 1998 conjectured that every graph $G$ satisfies $χ(G) \leq \lceil \frac{Δ(G)+1+ω(G)}{2} \rceil$. As a partial result, he proved the existence of $\varepsilon > 0$ for which every graph $G$ satisfies $χ(G) \leq \lceil (1-\varepsilon)(Δ(G)+1)+\varepsilonω(G) \rceil$. We propose an analogue conjecture for digraphs. Given a digraph $D$, we denote by $\vecχ(D)$ the dichromatic number of $D$, which is the minimum number of colours needed to partition $D$ into acyclic induced subdigraphs. We let $\overleftrightarrowω(D)$ denote the size of the largest biclique (a set of vertices inducing a complete digraph) of $D$ and $\tildeΔ(D) = \max_{v\in V(D)} \sqrt{d^+(v) \cdot d^-(v)}$. We conjecture that every digraph $D$ satisfies $\vecχ(D) \leq \lceil \frac{\tildeΔ(D)+1+\overleftrightarrowω(D)}{2} \rceil$, which if true implies Reed's conjecture. As a partial result, we prove the existence of $\varepsilon >0$ for which every digraph $D$ satisfies $\vecχ(D) \leq \lceil (1-\varepsilon)(\tildeΔ(D)+1)+\varepsilon\overleftrightarrowω(D) \rceil$. This implies both Reed's result and an independent result of Harutyunyan and Mohar for oriented graphs. To obtain this upper bound on $\vecχ$, we prove that every digraph $D$ with $\overleftrightarrowω(D) > \frac{2}{3}(Δ_{\max}(D)+1)$, where $Δ_{\max}(D) = \max_{v\in V(D)} \max(d^+(v),d^-(v))$, admits an acyclic set of vertices intersecting each biclique of $D$, which generalises a result of King. We finally give a short proof that all oriented graphs $D$ satisfy $\vecχ(D) \leq \frac{\sqrt{2}}{2} \tildeΔ(D) + 2$, improving on a result of Golowich.

math.CO↗
arXiv2026

Parameterised algorithms for temporally satisfying reconfiguration problems

Given a static vertex-selection problem (e.g. independent set, dominating set) on a graph, we can define a corresponding temporally satisfying reconfiguration problem on a temporal graph which asks for a sequence of solutions to the vertex-selection problem at each time such that we can reconfigure from one solution to the next. We can think of each solution in the sequence as a set of vertices with tokens placed on them; our reconfiguration model allows us to slide tokens along active edges of a temporal graph at each time-step. We show that it is possible to efficiently check whether one solution can be reconfigured to another, and show that approximation results on the static vertex-selection problem can be adapted with a lifetime factor to the reconfiguration version. Our main contributions are fixed-parameter tractable algorithms with respect to: enumeration time of the related static problem; the combination of temporal neighbourhood diversity and lifetime of the input temporal graph; and the combination of lifetime and treewidth of the footprint graph.

cs.DS↗
arXiv2026

Oracle-free Boltzmann Sampling for Powersets

We propose an approach for sampling powersets under the Boltzmann distribution in an oracle-free way, i.e. without numerically evaluating the associated generating function. Our approach relies on a Poissonised infinite occupancy model and thinning. It yields an explicit sampler for bounded counting sequences and extends under mild growth conditions. We implement the sampler and find runtimes comparable to existing Boltzmann samplers.

cs.DM↗
arXiv2026

Extending Exact Integrality Gap Computations for the Metric TSP

The subtour relaxation of the traveling salesman problem (TSP) plays a central role in approximation algorithms and polyhedral studies of the TSP. A long-standing conjecture asserts that the integrality gap of the subtour relaxation for the metric TSP is exactly $4/3$. In this paper, we extend the exact verification of this conjecture to a larger number of vertices. Using the framework introduced by Benoit and Boyd in 2008, we confirm their results up to $n=10$. We further show that for $n=11$ and $n=12$, the published lists of extreme points of the subtour polytope are incomplete: one extreme point is missing for $n=11$ and twenty-two extreme points are missing for $n=12$. We extend the enumeration of the extreme points of the subtour polytope to instances with up to $n=16$ vertices in the general case. Restricted to half-integral extreme points, we extend the enumeration to $n=18$. Our results provide additional support for the $4/3$-Conjecture. Our lists of extreme points are available on the public bonndata repository (https://doi.org/10.60507/FK2/JK95PC).

math.CO↗
arXiv2026

Connectivity of Districting Metagraphs

In this article, we prove irreducibility results for a family of Markov chains arising in the study of redistricting and detecting gerrymandering. These chains use ReCom moves as their transition mechanism and are commonly employed in Markov chain Monte Carlo methods to generate ensembles of districting plans. Such ensembles are frequently used for outlier analysis, in which a proposed districting map is compared against the ensemble to determine whether it behaves atypically; this methodology often appears in expert testimony in redistricting litigation. We show that when the underlying dual graph is a triangular subset of the triangular lattice with side length 5 or larger, and each district consists of two merged geographic regions, the associated ReCom chain is irreducible. This provides another entry in the very small list of known classes of ReCom chains for which irreducibility has been established. We also demonstrate the fragility of this phenomenon by constructing an infinite family of maps for which the corresponding ReCom chain is not irreducible. Indeed, we produce a districting map that, after implementing a single ReCom move, always yields the same original map. These examples remain structurally close to the triangular lattice: they arise as subdivisions of the triangular lattice, and the resulting graphs have maximum degree at most 8. Finally, we prove irreducibility for a further special case: the ReCom chain on a 3 x n grid graph partitioned into three districts of size n.

math.CO↗
arXiv2026

A linear bound for nested cycles without geometric crossings

Cycles $C_1,\ldots,C_k$ in a graph are called nested without geometric crossings if they are pairwise edge-disjoint, $V(C_k)\subseteq\cdots\subseteq V(C_1)$, and each pair of consecutive cycles induces the same cyclic order on the vertices of the inner cycle, up to reversal. Let $f_k(n)$ be the least number of edges that forces such a family in every $n$-vertex graph. Gil Fernández, Kim, Kim and Liu proved that $f_2(n)=O(n)$, answering a question of Erdős, and asked whether $f_k(n)=O_k(n)$ for every fixed $k$. We prove this for all $k$. The proof selects the inner cycles together with a disjoint subgraph that supplies their external neighbours. A reselection argument gives disjoint paths from every inner-cycle vertex to any sufficiently large target set. Sublinear expansion and a rooted clique minor then allow the vertices to be joined in the required cyclic order.

math.CO↗
arXiv2026

A Note on Sphere Packing Bounds for Tuple Lattice Sieving

A finite set of unit vectors is $k$-irreducible if every signed sum of between two and $k$ distinct elements has norm greater than one. Let $\mathcal{R}_k$ be the maximal asymptotic rate of such sets, and let $κ(α)$ be the maximal asymptotic rate of spherical codes with pairwise inner products at most $α$. For $k \ge 2$ we show: \begin{align} \mathcal{R}_k \le \min_{1 \le r \le \lfloor k/2 \rfloor} \frac{1}{r} \, κ\!\left(1 - \frac{1}{2r}\right) \, . \end{align} Combining this with standard sphere packing bounds, for large $k$ we obtain an almost-tight asymptotic comparison with the known lower bounds: \begin{align} \left(\tfrac{1}{2}-o(1)\right) \, \frac{\log_2 k}{k} \le \mathcal{R}_k \le (1 + o(1)) \, \frac{\log_2 k}{k} \, . \end{align}

math.CO↗
arXiv2026

An Elementary Proof of the $\widetilde O(n^{1/3})$ Bound for Separating Words

For two distinct binary words of length $n$, the separating words problem asks for a small deterministic finite automaton that accepts exactly one of them. Chase proved a $\widetilde O(n^{1/3})$ upper bound using a complex-analytic estimate for sparse polynomials. We replace that estimate by a finite-difference argument and a second-order real recurrence cutoff. The resulting elementary proof gives an explicit bound of $O(n^{1/3}(\log n)^{7/3})$ states.

cs.FL↗
arXiv2026

Graph-based automata

We study graph-based automata: nondeterministic finite automata obtained from edge-colored or oriented graphs by taking every vertex as both initial and accepting, and every edge as a pair of opposite transitions. The language of these automata corresponds to the set of edge-colored or oriented paths mapping to their corresponding graphs. We develop an analogous notion for trees and characterise the languages recognised by these models. For tree languages we prove the existence of a unique size- minimal graph and, more generally, a homomorphism-minimal graph for both word and tree languages using duality methods. In order to further motivate these models, we showcase a few results at the inter- section between graph theory and automata theory: We relate graph-based automata to reversible automata, give a decomposition of graph-based languages into reversible languages, and introduce the remanent language of an undirected graph as the inter- section of all its orientations. This remanent language captures structural information on the graph such as chromatic number.

cs.FL↗
arXiv2026

Degenerating orbits of the Longest Edge Bisection process

We study the Longest Edge Bisection (LEB) process as a dynamical system on the projective shape space of simplices. A long-standing conjecture going back to Adler and Rivara-Levin and motivated by finite-element mesh refinement, often taken as a standing assumption, is that this procedure is non-degenerate and, in fact, in a certain way periodic. We prove: \begin{itemize} \item There are 3-dimensional simplices such that the longest edge-bisection algorithm degenerates. \item There is an open set of 4-dimensional simplices on which the longest edge-bisection algorithm degenerates. \item If parametrizing the space of $d$-dimensional simplices by independent standard Gaussian vectors, then as $d$ increases, a random simplex degenerates asymptotically almost surely. \end{itemize} This is realized through exhibiting hyperbolic behaviour of the LEB process. We also exhibit elliptic behaviour that is nonperiodic.

math.DS↗
arXiv2026

A $(\log n)^{1/4}$ Bound for the Komlós Problem

Let $A\in\mathbb{R}^{m\times n}$ have columns of Euclidean norm at most one. We prove that $\operatorname{disc}(A)\le2395\left(1+\log_+\frac n9\right)^{1/4}+2\sqrt2$. Here $\log_+t=\max\{0,\log t\}$. Building on Bansal and Jiang's affine spectral independence framework, we remove the $(\log\log n)^{7/4}$ factor from their bound. The fourth root comes from balancing the logarithmic decrease in the alive dimension against the fourth power of the row thresholds. Historical exponential sums control the covariance budget across size classes with summable thresholds. An exact threshold-sum certificate gives the coefficient $2395$, and rounding at most eight remaining fractional coordinates costs $2\sqrt2$. The finite construction also gives partial colourings from any prescribed starting point and at any prescribed depth, preserving existing signs. We formalize the partial- and full-colouring theorems in Lean, including the finite trajectory, exact threshold sum and final rounding, with Bansal--Jiang Theorem A.4 as the sole external research theorem assumption.

math.CO↗
arXiv2026

Abelian Cayley High-Dimensional Expanders with Polylogarithmic Degree

We construct an explicit infinite family of simple two-dimensional Cayley complexes over $\mathbb{F}_2^n$ whose degree is polynomial in $n$ and whose nontrivial vertex-link eigenvalues lie in $[-λ,λ]$ for every fixed $λ>0$. For every fixed $d\ge2$, we also obtain an explicit infinite family of weighted $d$-dimensional Cayley complexes over $\mathbb{F}_2^n$ with codimension-two local spectral norm at most $1/d$ and Cayley degree $Θ_d(n)$. Our two-dimensional construction uses evaluation at rational points of algebraic curves to produce projective direction sets and many functions affine along these directions, which may be useful for further constructions and improvements.

math.CO↗
arXiv2026

Random Schreier graphs as expanders

Expander graphs, due to their mixing properties, are useful in many algorithms and combinatorial constructions. One can produce an expander graph with high probability by taking a random graph (e.g., the union of $d$ random bijections for a bipartite graph of degree $d$). This construction is much simpler than all known explicit constructions of expanders and gives graphs with good mixing properties (small second largest eigenvalue) with high probability. However, from the practical viewpoint, it uses too many random bits. The natural idea is to restrict the class of the bijections that we use. For example, if both sides are vector spaces $\mathbb{F}_q^k$ over a finite field $\mathbb{F}_q$, we may consider only \emph{linear} bijections, making the number of random bits polynomial in $k$ (and not $q^k$). In this paper we provide some experimental data that shows that this approach conserves the mixing properties (the second eigenvalue) for several types of graphs (undirected regular and biregular bipartite graphs). We also prove some upper bounds for the second eigenvalue (though they are quite weak compared with the experimental results) and a lower bound for a special class of graphs. Finally, we discuss the possibility to decrease the number of random bits further by using Toeplitz matrices; our experiments show that this change makes the mixing properties only marginally worse while the number of random bits decreases significantly.

math.CO↗
arXiv2026

Maximum-Entropy Random Walks on Hypergraphs

Random walks are fundamental tools for analyzing complex networked systems, including social networks, biological systems, and communication infrastructures. While classical random walks focus on pairwise interactions, many real-world systems exhibit higher-order interactions naturally modeled by hypergraphs. Existing random walk models on hypergraphs often focus on undirected structures or do not incorporate entropy-based inference, limiting their ability to capture directional flows, uncertainty, or information diffusion in complex systems. In this article, we develop a maximum-entropy random walk framework on directed hypergraphs with two interaction mechanisms: broadcasting where a pivot node activates multiple receiver nodes and merging where multiple pivot nodes jointly influence a receiver node. We infer a transition kernel via a Kullback--Leibler divergence projection onto constraints enforcing stochasticity and stationarity. The resulting optimality conditions yield a multiplicative scaling form, implemented using Sinkhorn--Schrödinger-type iterations with tensor contractions. We further analyze ergodicity, including projected linear kernels for broadcasting and tensor spectral criteria for polynomial dynamics in merging. The effectiveness of our framework is demonstrated with both synthetic and real-world examples.

eess.SY↗
arXiv2026

List coloring uncrowded hypergraphs at the shattering threshold

Improving an earlier bound of Frieze and Mubayi, Iliopoulos showed that any $k$-uniform uncrowded hypergraph of maximum degree $Δ$ has list chromatic number at most $(1+o(1))(k-1)\Big(\fracΔ{\logΔ}\Big)^{\frac{1}{k-1}}$. Determining the optimal leading constant in this bound remains a major open problem. A natural target for the constant, suggested by the coupon-collector heuristic, is $(k-1)^{\frac{1}{k-1}}$, which also arises independently as the shattering threshold for coloring and finding large independent sets in sparse random hypergraphs. We show that uncrowded hypergraphs attain this threshold: for $k \ge 2$, every $k$-uniform uncrowded hypergraph $\mathcal{H}$ of maximum degree at most $Δ$ satisfies $$χ_\ell(\mathcal{H})\le(1+o(1))\Big((k-1)\fracΔ{\logΔ}\Big)^{\frac{1}{k-1}}.$$ This matches the coupon-collector heuristic, and establishes a 2019 conjecture of Molloy for uncrowded hypergraphs. As a consequence, we show that every $k$-uniform linear hypergraph of maximum degree at most $Δ$ has chromatic number at most $c_k\Big(\fracΔ{\log Δ}\Big)^{\frac{1}{k-1}}$ with $c_k=1-o_k(1)$, resolving, in a stronger form, a recent conjecture of Verstraëte and Wilson on the independence number of linear hypergraphs. Our techniques yield the first palette sparsification result for hypergraph coloring using $o(Δ^{\frac{1}{k-1}})$ colors per vertex. Specifically, for an uncrowded $k$-graph $\mathcal{H}$ and $q=Θ\Big({\Big(\fracΔ{γ\logΔ}\Big)^{\frac{1}{k-1}}}\Big)$, sampling $Θ(Δ^{\fracγ{k-1}})$ colors per vertex suffices to obtain a proper $q$-coloring of $\mathcal{H}$ w.h.p. As an algorithmic consequence, we obtain a single-pass streaming algorithm using $O\big(n^{1+o(1)}\big)$ space for uncrowded $k$-graphs, and, via hypergraph partitioning, for linear hypergraphs as well.

math.CO↗
arXiv2026

Resolving a conjecture on quadratic APN functions and a new quadratic $(n,n)$-function associated to crooked functions

We say an $(n,n)$-function $F \colon \mathbb{F}_2^n \to \mathbb{F}_2^n$ is a crooked function if for any nonzero $a \in \mathbb{F}_2^n$, the image of $D_aF(x)=F(x)+F(x+a)$ is an affine hyperplane. The only known examples of crooked functions are all quadratic almost perfect nonlinear (APN), or equivalently, for every known crooked function, $D_aF$ is affine for all $a \in \mathbb{F}_2^n$. The ortho-derivative $π_F \colon\mathbb{F}_2^n \to \mathbb{F}_2^n$ of a crooked function $F$ is the function such that $π_F(0)=0$, and for any nonzero $a$, the set $\{0,π_F(a)\}^\perp$ is the underlying vector space of $\mathrm{Im}(D_aF)$. We prove that for $n \geq 4$ and a crooked function $F$, if $k$ is a non-negative integer such that $F$ has $2^k-1$ quadratic component functions, $π_F$ has at least $2^n-2^{n-k}$ component functions of algebraic degree $n-2$. In particular, we resolve Gorodilova's conjecture that every component function of $π_F$ has algebraic degree $n-2$ when $F$ is quadratic APN. As a second main result, for $n \geq 4$, we associate to a crooked function $F$ a quadratic function $\varepsilon_F \colon \mathbb{F}_2^n \to \mathbb{F}_2^n$ that satisfies a strong geometric-combinatorial condition regarding the sums of $F$ over $2$-dimensional linear subspaces. As a corollary to both of our main results, we prove that for any even $n \geq 4$, any quadratic APN $(n,n)$-function has at least $n$ semi-bent components. Furthermore, we obtain a congruence result on a problem on $m$-sequences introduced by Johansen, Helleseth, and Kholosha, and we determine the exact algebraic degrees of some Boolean functions associated to the bent and near-bent components of particular classes of plateaued vectorial functions.

math.CO↗
arXiv2026

Tensor network representations of discrete maximum entropy distributions via mean polytopes

We present tensor network representations for discrete maximum entropy distributions under expectation constraints. To this end, we introduce Computation-Activation Networks (CompActNets), a tensor network architecture that subsumes exponential families. By leveraging the geometry of the convex polytope of realizable expectation vectors, we represent any maximum entropy distribution in the same architecture. We exploit the fact that proper faces of this polytope correspond to the boundary closure of exponential families, which restricts the distribution's support. We then derive explicit representations for the support within the CompActNet architecture. The proposed framework suggests tensor network ranks as complexity measures for faces. Finally, a case study on Boolean statistics links the geometry of 0/1-polytopes directly to propositional formulas.

math.ST↗
arXiv2026

A simple algorithm for computing Hamilton paths on independent set polytopes

The independent set polytope, or stable set polytope, of a graph $G$ is the 0/1-polytope defined by the convex hull of the characteristic vectors of all independent sets of $G$. We present a simple algorithm for computing a Hamilton path on the independent set polytope of a given $n$-vertex graph $G$ with amortized delay $\mathcal{O}(n)$. The independent sets are listed such that two consecutive sets differ either in removing a vertex, or adding a vertex and removing its neighbors from the independent set, i.e., the symmetric difference between two consecutive independent sets induces a star in $G$. As applications of this result, we obtain an algorithm to compute a Hamilton path on the matching polytope of an $m$-edge graph $G$ with worst-case delay $\mathcal{O}(m)$, which lists all matchings of $G$ in such a way that the symmetric difference between two consecutive matchings is a path on at most three edges. Furthermore, we obtain an algorithm to compute a Hamilton path on the chain polytope and order polytope of an $n$-element poset $P$ with amortized delay $\mathcal{O}(n)$, which lists all antichains of $P$ or all ideals of $P$, respectively, by star exchanges. Our algorithms are derived from the generic framework proposed by Merino and Mütze (FOCS'23+SICOMP'24) for computing Hamilton paths on arbitrary 0/1-polytopes, which uses a linear optimization procedure as a black box. Our algorithms bypass solving the computationally intractable maximum weight independent set problem by a simple and purely combinatorial greedy rule.

cs.DM↗
Compare source metadata on this page
WorkPublishedSource identifierSource
An analogue of Reed's conjecture for digraphs2026-09-082407.05827arxiv
Parameterised algorithms for temporally satisfying reconfiguration problems2026-09-082502.11961arxiv
Oracle-free Boltzmann Sampling for Powersets2026-09-082601.09508arxiv
Extending Exact Integrality Gap Computations for the Metric TSP2026-09-082603.12995arxiv
Connectivity of Districting Metagraphs2026-09-082606.10152arxiv
A linear bound for nested cycles without geometric crossings2026-09-082609.02234arxiv
A Note on Sphere Packing Bounds for Tuple Lattice Sieving2026-09-082609.08190arxiv
An Elementary Proof of the $\widetilde O(n^{1/3})$ Bound for Separating Words2026-09-082609.08191arxiv
Graph-based automata2026-09-082609.08843arxiv
Degenerating orbits of the Longest Edge Bisection process2026-09-082609.08846arxiv
A $(\log n)^{1/4}$ Bound for the Komlós Problem2026-09-082609.08885arxiv
Abelian Cayley High-Dimensional Expanders with Polylogarithmic Degree2026-09-082609.08937arxiv
Random Schreier graphs as expanders2026-09-072305.02154arxiv
Maximum-Entropy Random Walks on Hypergraphs2026-09-072603.12098arxiv
List coloring uncrowded hypergraphs at the shattering threshold2026-09-072606.18048arxiv
Resolving a conjecture on quadratic APN functions and a new quadratic $(n,n)$-function associated to crooked functions2026-09-072608.23888arxiv
Tensor network representations of discrete maximum entropy distributions via mean polytopes2026-09-072609.07184arxiv
A simple algorithm for computing Hamilton paths on independent set polytopes2026-10-092609.07304arxiv

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