Papers

Motivated by the structure of various standard statistical arrays, Bailey, Praeger, Rowley and Speed (1983) described the automorphism groups of poset block structures as generalized wreath products. We observe that these are exactly the groups used to define DAG exchangeability in Jung, Lee, Staton and Yang (2021), and we restate the representation theorem of the latter work in this language. As a concrete example, we write out the resulting representation for block matrices whose block rows and block columns are separately exchangeable, and whose rows and columns may also be permuted independently inside each block.

For a graph $F$, its extremal number ${\rm ex}(n,F)$ and extremal graphs denote the maximum number of edges in $F$-free graphs and the graphs attaining this maximum, respectively. Let $h_F(n,q)$ be the minimum number of copies of $F$ in an $n$-vertex graph with ${\rm ex}(n,F)+q$ edges, and let $t_F(n,q)$ denote the corresponding minimum obtained by adding $q$ edges to an extremal graph. Mubayi (2013) conjectured that for every non-$r$-partite stable $r$-uniform hypergraph $F$ with $r\ge 2$, $h_F(n,q)\ge q\cdot t_F(n,1)$ holds for all sufficiently large $n$ and all integers $q\ge 1$. In this paper, we prove that Mubayi's conjecture fails in a stronger sense. For every integer $d\ge 2$, we construct a family of non-$d$-partite stable graphs $F$ such that $h_F(n,q)< q\cdot t_F(n,1)$ for all $1\le q\le O\bigl(n^{4/ 3}\bigr)$. In particular, these examplesis completely refute Mubayi's conjecture. \noindentKey words: extremal graph; supersaturation; chromatic number.

We construct the Conway-Wales lattice from a self-dual even code $C$ over $\mathbb{Z}_4$ whose binary layers interweave the Hamming codes $\mathcal{H}_7$ and $\mathcal{H}_8$. Explicit signed symmetries lift every automorphism of the binary residue, and the full signed code group is the coordinate-frame stabilizer of the Gaussian lattice. An integral reflection in an embedded scaled self-dual sublattice supplies an additional symmetry, from which the classical frame exchange is recovered. Counting intrinsically defined frames determines the full Gaussian isometry group, of order $583\,704\,576\,000$. Its quotient by the four Gaussian scalars is the Rudvalis simple group, and the derived subgroup of a cross stabilizer is the Tits simple group of order $17\,971\,200$. The construction, automorphism-group determination, and both simplicity proofs are carried out within the lattice, without assuming the known group orders or using a recognition or classification theorem.

Let $G=\langle a,b\rangle$ be a finite group with distinct nonidentity generators, and suppose that $H=\langle ba^{-1}\rangle$ is normal. We determine exactly which complementary generator choices on the nonterminal cosets of $H$ extend to two arc-disjoint Hamiltonian paths with both terminal vertices in $H$. The associated return residue must avoid an explicit set of at most six residues. When $[G:H]\ge3$, any two nonterminal cosets may be left free while all other choices are prescribed; two is best possible. Every such Cayley digraph has two arc-disjoint Hamiltonian paths; this includes all two-generator Cayley digraphs of finite abelian groups. The criterion follows from a complete distance formula for the Hamiltonian cut spectra of $\mathrm{Cay}(\mathbb{Z}_q;\{D,D+1\})$. Their unique nondegenerate exception comes from the odd central coefficient $(q-1)/2$. The geometric ingredient is a confinement theorem: two sector-filled integer sequences satisfying a length inequality, with overlapping ranges and distance at least two, have one sequence contained in a single gap of the other. Sector filling bounds alternation, and the length inequality excludes interlacing. A self-contained proof of the primitive-ray cycle profile follows the intersection geometry of Curran and Witte.

Let $m,n \geq 3$. A set of vertices $S$ of the rectangular grid graph $P_m \square P_n$ is resolving if the taxicab distance vectors of the vertices of $P_m \square P_n$ with respect to $S$ are pairwise distinct. A resolving set $S$ is minimal if no proper subset of $S$ is resolving, and a minimal resolving set of cardinality $k$ is called a $k$-minimal. We extend the work of Andersen et al., who characterized $3$-minimals, established the maximum cardinality $2\min(m,n)-2$, and posed the complete characterization and enumeration of minimal resolving sets for grids as an open problem, and Adar and Epstein, who showed that $3$ is the only possible odd cardinality and that every minimal resolving set of cardinality at least $4$ can be ordered to form a sequence corresponding to a zigzag sequence. We provide a recursive construction that generates exactly the minimal resolving sets of cardinality at least $4$ for grids. We derive from the construction closed-form formulas to enumerate the $k$-minimals for all even $4 \leq k \leq 2\min(m,n) -2$. Together with the known characterizations for cardinalities $2$ and $3$ and a direct enumeration of the $3$-minimals, this yields a complete characterization and enumeration of the minimal resolving sets of rectangular grid graphs of every possible cardinality.

A strong edge colouring of a graph $G$ is an edge colouring in which every colour class is an induced matching. The minimum number of colours is the strong chromatic index $\chi'_s(G)$. If each edge $e$ is assigned a list $L'(e)$ and its colour must belong to $L'(e)$, the corresponding parameter is the strong list chromatic index $\chi'_{s,\ell}(G)$. From the definitions, $\chi'_s(G)\le\chi'_{s,\ell}(G)$. Barrett et al. gave an $8$-approximation for strong edge colouring on unit disk graphs and Grelier et al. improved the approximation factor to $6$. Our first result extends this factor-$6$ guarantee from unit disk graphs to the strictly larger class of disk graphs. In another direction, Erdős and Nešetřil conjectured that the strong chromatic index of a graph of maximum degree $\Delta$ is asymptotically at most $1.25\Delta^2$. The best published general asymptotic upper bound has leading coefficient $1.772$, due to Hurley et al. For unit disk graphs, Dębski et al. proved that $\chi'_s(G) \leq 1.625 \Delta^2$. Our second result improves this leading coefficient to $225/142 \approx 1.5845$. In fact, the proof establishes a stronger bound $\chi'_{s,\ell}(G)\le\frac{225}{142} \Delta^2+O(\Delta)$ for unit disk graphs.

We generalize a construction of Herzog and Takayama pertaining to differential graded algebra structures on mapping cones. Via this generalization, we provide the minimal free resolution of the edge ideal of a new graph built from suspension over a vertex cover of an old graph, and we discuss when this process preserves differential graded algebra structure on the respective resolutions.

In this paper, we study the higher-order ranking, by spanning-tree count, of graphs obtained from a complete graph by deleting a fixed number of edges. Using the edge-defect matrix determined by the deleted edges and the interaction number measuring the local overlap among them, we derive a stability inequality that quantitatively estimates the decrease in the number of spanning trees from the matching-deletion case. Combining this stability estimate with a classification of deletion graphs having small interaction number, we determine, up to isomorphism, the nine deletion graphs with the largest spanning-tree counts for $p\geq6$ and $n\geq2p$. We also clarify the relation between the interaction number, the local structure of the deletion graph, and the decrease in the number of spanning trees through a logarithmic expansion of the normalized spanning-tree count.

A famous folklore result in list coloring demonstrating that the gap between the list chromatic number and chromatic number of a graph can be arbitrarily large is: $\chi_{\ell}(K_{l,t}) = 1+l$ if and only if $t \geq l^l$. DP-coloring (also called correspondence coloring) is a well-studied generalization of list coloring introduced in 2015. In 2018, Mudrock studied the DP analogue of the aforementioned folklore result. He proved that for $l \in\mathbb{N}$, if $\mu(l)$ is the smallest integer $t$ such that $\chi_{DP}(K_{l,t})=1+l$, then $\left\lceil l^l/l!\right\rceil \leq \mu(l) \leq 1+l^l(\log(l!)+1)/l!$. Recently, Kaul, Mudrock, and Sharma studied a more general version of this problem by studying the smallest $t$ for which $\chi_{DP}(G \square K_{l,t}) = k + l$, where $G$ satisfies certain criticality conditions and $G \square K_{l,t}$ denotes the Cartesian product of $G$ and $K_{l,t}$. In this paper, we introduce a notion we call covering families that gives a new perspective on these DP-coloring questions. In particular, if $\kappa(l)$ denotes the minimum size of a covering family of $[l]^l$, we show that $\mu(l)=\kappa(l)$. We use this equivalence to prove $\mu(4)=12$ and to obtain new general lower bounds on $\mu(l)$. We also prove a general upper bound on the minimum size of covering families which yields an improved general upper bound on $\mu(l)$ and gives improvements on known bounds for related DP-coloring questions involving Cartesian products with complete bipartite graphs.

Let $G$ be a graph. Under a strong edge coloring of $G$, every color class is an induced matching. The strong chromatic index of $G$, denoted by $\chi'_s(G)$, is the smallest integer $k$ such that $G$ admits a strong edge coloring with $k$ colors. Denote by $\Delta(G)$ the maximum degree of $G$. In this paper, we prove that every graph $G$ with $\Delta(G)\le 6$ satisfies $\chi'_s(G)\le 57$, improving the best known upper bound $60$.

Every group $G$, of arbitrary cardinality, admits a right-invariant metric with at most five values whose isometry group is exactly the group $\Xi(G)$ of permutations preserving every inverse-pair difference. Five is sharp. Writing $N(G)$ for the least number of values, including zero, we obtain $N(G)=1$ for the trivial group, $N(G)=2$ for $C_2$ and $C_3$, $N(G)=3$ for every other group outside an explicitly listed set $\mathcal{E}$ of 24 finite groups, $N(G)=4$ on $\mathcal{E}\setminus\{C_2^3, C_3^2\}$, and $N(C_2^3)=N(C_3^2)=5$. The exact result combines Cayley-graph realization theorems with verified finite certificates and hand proofs of the lower bounds. Separately, a uniform constructive 32-value bound uses subgroup-entry ranks, sign-variation conflict labels, and short-word rigidity, but no Cayley-index input or computation. In Knill's right-translation sense, a right-invariant metric on $G$ is naturalizing when every group law on its underlying set with isometric right translations is isomorphic to $G$; a group is natural when it admits such a metric. We prove that an abelian group $A$ is natural if and only if $2A=A$ or $2A=\{0\}$, and a nonabelian group is natural if and only if it is not generalized dicyclic. Every natural group admits a naturalizing metric with at most five values, and this bound is sharp.

A dominating set $D$ of a nontrivial connected graph $G$ is called a semitotal dominating set of $G$ if every vertex in $D$ is at distance at most two from another vertex in $D$. If, in addition, $D$ is an independent set, then $D$ is called an independent semitotal dominating set of $G$. The (independent) semitotal domination number of $G$ is the minimum cardinality among all (independent) semitotal dominating sets of $G$. In this paper, we obtain closed formulas for these parameters in the following three well-known graph operators defined from a connected graph: the subdivision, middle, and central graphs.

We characterize uniform factor-balancedness in Arnoux-Rauzy words, generalizing the work of Espinoza, Popoli, and Stipulanti on ternary alphabets to finite alphabets of any size. In particular, we prove that an Arnoux-Rauzy word $\mathbf{x}$ is uniformly factor-balanced if and only if $\mathbf{x}$ has bounded weak partial quotients.

A graph $G$ is det-extremal if $|\operatorname{det} A|=\operatorname{per} A$ for its adjacency matrix $A$. Det-extremal cubic bipartite graphs arise in the study of Pólya's permanent problem, and McCuaig characterized the $3$-connected ones as vertex-sums of copies of the Heawood graph. The total domatic number of a graph is the largest number of pairwise disjoint total dominating sets. Characterization of the cubic graphs with total domatic number $1$ has been a long-standing open problem. In this paper, we prove that a connected cubic graph is det-extremal if and only if its total domatic number is $1$. We further show that McCuaig's characterization extends to all $3$-connected cubic graphs, and that every connected det-extremal cubic graph has girth $3$, $5$ or $6$. We also prove that a connected det-extremal cubic non-bipartite graph has at least $28$ vertices, and that this bound is best possible. Through this correspondence, these results carry over to cubic graphs with total domatic number $1$. In addition, in the language of configurations, our results imply that every triangle-free $3$-configuration has a blocking set.

We show that the percolation components on a square grid can be $5$-colored as a factor of iid so that neighboring components get different colors. Above the critical probability, we observe that these regions can be 4-colored. Along the way, we extend the classical correspondence between fiid process and measurable labellings of the Bernoulli shift to this quotient setting, prove a general coloring result about hyperfinite pmp planar graphs, and clarify some foundational issues about how to define planarity for Borel graphs.

A unitary signing of a $d$-regular graph is a Hermitian adjacency matrix $M$ whose nonzero entries lie in $\{\pm1,\pm i\}$ and satisfies $M^2=dI$. Motivated by the work of Alon and Zheng on orthogonal and unitary signings of cube-like graphs, we introduce the $\Theta$-property for a generating set $S\subseteq\mathbb Z_2^n$: whenever three pairwise disjoint subsets of $S$ have the same sum, at least two of them have even size. We prove that every zero-free generating set with the $\Theta$-property gives a cube-like graph $Q_S$ admitting a unitary signing, and we give an explicit local formula for such a signing. For Sidon sets, the $\Theta$-property is also necessary, yielding a characterization of the Sidon cube-like graphs that admit unitary signings. In this setting there are exactly $2^{|S|-n}$ switching-equivalence classes of unitary signings, and we characterize when a unitary signing can be chosen to be orthogonal. The $\Theta$-property admits a linear-algebraic description in terms of the dependency space $\mathcal D$ of $S$: \[ |D_1\cap D_2| \equiv |D_1||D_2| \pmod2 \qquad (D_1,D_2\in\mathcal D). \] Equivalently, the map \[ D\longmapsto \binom{|D|}{2}\pmod2 \] is linear on $\mathcal D$. For the corresponding extremal set problem, in which zero is permitted, this formulation yields the sharp bound $|S|\leq 2n+1$ for generating sets with the $\Theta$-property. We give elementary constructions attaining this bound for every $n\geq3$. Under the additional Sidon condition, the same extremal value is attained for every $n\geq10$ using binary self-dual codes. The exact Sidon maximum is also determined for $3\leq n\leq9$.

We investigate arithmetic and combinatorial properties of the function $a(n)$, which is the signed number of partitions of $n$ into exactly two distinct part sizes, each occurring an odd number of times. We prove that $a(n)\ge 0$ for all $n$, establishing a positivity phenomenon for a signed partition function arising from a double Lambert series. Furthermore, we show that $a(n)$ satisfies nontrivial divisibility properties modulo $3$. The results reveal unexpected arithmetic regularity in a family of partition functions defined by odd multiplicity constraints and restricted support.

A $D$-diffsequence is a sequence of integers $x_1 < \dots < x_k$ such that $x_{i+1} -x_i \in D$ for $1 \le i \le k-1$. The set $D$ is called $r$-accessible if every $r$-coloring of the positive integers contains arbitrarily long monochromatic $D$-diffsequences. This note proves that the set of Fibonacci numbers is not 3-accessible, resolving an open problem of Landman and Robertson from 2007.

We prove an inverse theorem for the Gowers uniformity norm $\mathsf{U}^k(\mathbb{F}_p^n)$ with quasipolynomial bounds in the case when $p \geq k$. The inverse theorem follows from a quasipolynomial structure theorem for Freiman multihomomorphisms, which are a natural generalization of Freiman homomorphisms to maps of several variables. The proof of the structure theorem for Freiman multihomomorphisms is the central result of the paper and rests on three main ingredients: algebraic regularity method, abstract Balog-Szemerédi-Gowers theorem and the theory of multilinear maps defined on multilinear varieties. The last ingredient originates from an earlier work of Gowers and the author, and is significantly expanded in this paper. In particular, once the theory of such maps is in place, the proof of the structure theorem for Freiman multihomomorphisms is relatively short, especially compared to the previous quantitative results in the inverse theory of Gowers norms.

In this paper, three ($q$-)log-convexity or concavity properties related to the $d$-Hoggatt numbers are investigated. First, we show that the $d$-Hoggatt transformation preserves the log-convexity of a sequence. We then establish the strong $q$-log-concavity of a $q$-analog of $d$-Hoggatt numbers. Finally, we prove the core result of this work, namely, the sequence of $d$-Hoggatt polynomials is strongly $q$-log-convex, based on the theory of Schur functions.

The notion of a linear subspace code in a projective space was introduced by Braun, Etzion and Vardy (2013), who conjectured that a linear subspace code in the projective space has at most 2^n codewords. We resolve this conjecture. A novel character-theoretic argument is used to prove the conjecture and the proof is self-contained.

We introduce and study a new family of circulant digraphs associated with the cyclic group $\Z_N$, obtained by restricting admissible combinations of two generators $a$ and $b$ to three coordinate sectors. The resulting distance-like function differs from the standard directed distance in circulant digraphs and gives rise to new geometric and combinatorial phenomena. Using planar lattice representations and periodic tessellations, we analyze the growth of reachable sets and derive Moore-type upper bounds for the corresponding order/diameter problem. We construct explicit infinite families of three-quarters circulant structures with the prescribed diameter and provide lattice-based methods for determining admissible generator pairs. Separate constructions are obtained for even and odd diameters. In addition, computational experiments for small and moderate orders suggest improved families for even diameters and motivate a conjectural asymptotic formula for the maximum attainable order. The paper highlights the interplay between constrained lattice representations, periodic tilings, and extremal problems for circulant networks.

A class of graphs is monadically dependent if one cannot encode all graphs in coloured graphs from the class using a fixed first-order formula, and monadically stable if one cannot even encode arbitrarily long linear orders. Bonnet et al. (ICALP 2025) characterised monadic dependence by flip-separability: for every vertex weighting, boundedly many flips - complementations of the adjacency relation within a vertex subset - make every ball of radius $r$ carry at most an $\varepsilon$-fraction of the weight, so that every set carrying an $\varepsilon$-fraction has two elements pulled apart. We introduce flip-packability: boundedly many graphs, each obtained from the input by boundedly many flips and all determined by the weighting before any set is presented, such that every set carrying an $\varepsilon$-fraction of the weight has $m$ elements pairwise far apart in one of them. The number of flips producing each graph depends on the radius alone; only the number of graphs depends on $\varepsilon$ and $m$. We prove that a class of graphs is flip-packable if and only if it is monadically stable, and $2$-flip-packable, that is, flip-packable with $m=2$, if and only if it is monadically dependent. The passage from two scattered elements to $m$ is thus exactly what separates the two notions. For monadically stable classes we show that the flipped graphs can be computed from the weighting in cubic time. Varying the three parameters of the definition - the sparsifying operation, the radius, and the number $m$ of elements scattered - produces eight known characterisations of sparse and dense graph classes from the same template. In each case $m$ separates a depth-like notion from its width-like relaxation: treedepth from treewidth, shrubdepth from cliquewidth, and monadic stability from monadic dependence.

For $d\ge2$ the $q$-deformed modular group specialized at a primitive $d$-th root of unity is the triangle group of type $(2,3,d)$. Using this we determine the fractions $r/s$ for which the $d$-th cyclotomic polynomial divides the denominator $S_{r/s}(q)$ of the $q$-deformed rational number $[r/s]_q$. They form the orbit of $\infty$ under the normal closure of the translation $z\mapsto z+d$ in $\mathrm{PSL}(2,\mathbb{Z})$. This proves a conjecture of Byakuno, Ren and Yanagawa, and shows that the congruences $s\equiv0$ and $r\equiv\pm1$ modulo $d$ characterize the divisibility exactly for $d\le5$. For $a\in\{2,3,4,6\}$ and $n>5a^2$ prime to $a$ we show that $S_{a/n}(q)$ is irreducible up to cyclotomic factors. Together with a computer check for small primes, this confirms a conjecture of Kogiso, Ren, Wakui, Yanagawa and the author for every prime $p$ and every $r$ prime to $p$ with $r\equiv\pm a$ or $ar\equiv\pm1\pmod p$.

For a connected graph \(G\), let \[ Q(G)=\frac{T(G;2,0)T(G;0,2)}{T(G;1,1)^2}. \] We obtain quantitative lower bounds for \(Q\) under the graph join operation. If \(A\) and \(B\) are arbitrary simple graphs of orders \(3\le a\le b\), then \(Q(A\vee B)\) admits an explicit lower bound depending only on \(a\) and \(b\), and this bound is strictly greater than \(1\). We further quantify the improvement produced by edges inside the two factors. For every simple graph \(F\), with \(n=|V(F)|+2\ge4\), we prove \[ Q(K_2\vee F)\ge \frac{27}{n^2}\left(\frac32\right)^{n-4}. \] Consequently, every join of at least three nonempty factors, and every complete multipartite graph with at least one edge and no cut edges, satisfies the strict multiplicative Merino–Welsh inequality. The proofs combine orientation estimates with spanning-tree comparisons based on effective resistance and block elimination.

We prove log-concavity for the determinant-weighted external semi-activity polynomials of all real flat arrangements, strengthening their known trapezoidality. In fact, we establish a quadratic coefficient inequality that, in rank at least two, implies power concavity with an explicit rank-dependent exponent. The proof uses a new mixed-volume representation of the coefficients and the Alexandrov–Fenchel inequality. A more general formula gives a factorization and log-concavity for related mixed-volume sequences. As applications, we establish the conjectured log-concavity for spanning-tree polynomials of Eulerian digraphs, extend it to positive circulation weights, and strengthen the coefficient inequalities for Alexander polynomials of special alternating links.

In an earlier paper, M. Talagrand made a number of daring conjectures. An AI model provided a very beautiful proof of one of them which we explain here.

We study the maximum sum rate of uniquely decodable code pairs. A weighted refinement of a complement-gluing construction yields an explicit code of length 672 and sum rate exceeding $1.318639029203$. An iterated projection argument, combined with a conditional entropy estimate and a justified continuous limit, gives an upper bound of $1.480063425539$.

For a nonempty acyclic oriented graph $H$, let $p(H)$ be the order of a longest directed path and let $\tau(H)$ be the least positive integer $n$ such that $H$ admits a homomorphism to every tournament of order $n$. For all $p\ge3$ and $g\ge1$, we construct a connected acyclic oriented graph $H$ with underlying girth greater than $g$, absolute and relative oriented clique numbers equal to three, and \[ p(H)=p,\qquad \tau(H)=r_{\mathrm{tr}}(p), \] where $r_{\mathrm{tr}}(p)=2^{\Theta(p)}$ is the tournament Ramsey number for a transitive $p$-vertex tournament. This disproves the conjectured polynomial bounds under bounded absolute or relative oriented clique number. It also shows that a forbidden graph can yield a polynomially $\tau$-bounded class only if its underlying graph is a forest. For fixed $g$, the least order of these examples is bounded by a polynomial in $p$. A separate construction gives maximum in- and outdegree $O(p^2)$, uniformly in $g$. For $p=4$, the least order is $2^{\Theta(g)}$. We also establish polynomial $\tau$-boundedness for every orientation of the two four-vertex trees. The pure-claw case follows from the known $O(p^4)$ bound. We obtain the bound $2p-2$ for mixed claws and one-turn orientations of $P_4$ when $p\ge2$, and bounds $4$ and $3p-2$ for the directed and alternating orientations of $P_4$, respectively. In the alternating case, $\tau(H)=p(H)$ when the underlying graph is triangle-free.

We develop a Hopf-algebraic theory of the graph coefficients arising in the quantum Magnus expansion. At the classical level, the graph expansion is governed by directed trees, whereas its quantum counterpart involves loop graphs, multiple edges, and different types of edge data. We introduce a contraction Hopf algebra of mixed quivers containing these physical diagrams and closed under contraction. On the Hopf subalgebra spanned by quivers, we define a character $e$ from normalized linear-extension data and a tadpole prescription, and let $\omega$ be its convolution inverse. Our main result is a universal closed formula for the connected function $\omega_c$ on every finite quiver. The formula is a finite sum over ordered set partitions of the vertex set and is valid without acyclicity or simplicity assumptions. We also give a local characterization: a two-vertex contraction identity, together with tadpole factorization, vanishing on disconnected graphs, and boundary data, determines $\omega_c$ uniquely. For acyclic quivers, the general formula reduces to a permutation formula whose coefficients depend only on descent numbers. We derive the same formula independently from the quantum Magnus expansion via operator products and Wick contractions, showing that its physical graph coefficients are governed by the convolution structure of the contraction Hopf algebra. As further consequences, we obtain a refined quantum Murua formula and orientation-sum identities related to Tutte and chromatic polynomials.