Papers

Let $\PG(2,q)$ denote the projective plane over the finite field $\mathbb{F}_q$ where $q\geq2$ is a prime power. For a positive integer $t$, let $B_{t}[\PG(2,q)]$ denote the $t$-page book obtained from $t-$many copies of the $(q+1)-$graph $\PG(2,q)$, sharing a common edge. Employing the $\mathsf{L}^p-$method and using the incidence structure of the projective plane, we establish that the upper bound of Turán density of $B_{t}[\PG(2,q)]$ is $(q(q+1)^{-1})^{\frac{1}{qt}}$. For the $(q+1)-$graph $\PG(2,q)$, the previously known upper bound on its Turán density was $1-\binom{q^2}{q}^{-1}$. As a consequence of our estimate, this upper bound is improved to $(q(q+1)^{-1})^{\frac{1}{q}}$.

We prove the 2143-3421 case of Burstein's conjecture on the distribution of descent-top sets by constructing an explicit bijection between the two avoidance classes. The bijection also preserves the pinnacle set and reverses the left-to-right order of the pinnacle values. We combine an involution on maximum Cartesian trees with exchanges of the final entries of maximal decreasing runs. On an intermediate avoidance class, we prove that the exchange procedures terminate and that the output in each direction is independent of the choices made. This gives an explicit inverse. Restricting the bijection to Dumont permutations of the first kind proves the conjecture of Burstein and Jones that 2143 and 3421 are Wilf-equivalent on this class

We give a randomized polynomial-time algorithm that colors any promised $3$-colorable graph on $n$ vertices with $\smash{O(n^{4/23}) = O(n^{0.17391\ldots})}$ colors, improving on the recent bounds of $O(n^{0.19539})$ by Bansal, Huang, and Lee and Narang and Tang who obtained $O(n^{(13-\sqrt{97})/18+\epsilon})=O(n^{0.17506\dots + \epsilon})$ colors for every fixed $\smash{\epsilon>0}$. To prove our result, we start from a fixed-level semidefinite relaxation, where we use a finite-depth recursion on Gaussian covers. Fixing a root vertex, we group vertices by correlation with the root vector. Here, each step extends a cover of directions by one edge and transfers it to a successor group. Our key analytic ingredient is a variance bound for Gaussian maxima: for a maximum of $m\geq 2$ centered linear forms with coefficient norms at most $r$, mean $\mu$, and variance $v$, we prove $v\leq r^2-\mu^2/(2\log m)$ using Chen's Gaussian convexity theorem. Together with a variance-scale lower-tail estimate, this controls the threshold loss at each extension, which shows that root-conditioned vector colorings can either extract a large independent set from a group or bound its size, forcing a contradiction after constantly many steps. The resulting sparse-case guarantee combines with the dense progress bound of Kawarabayashi, Thorup, and Yoneda, and the recursion's numerical inequalities are verified via rational interval arithmetic.

We give an explicit three-dimensional Hadamard matrix of order $22$. The example is presented by listing its twenty-two $22\times22$ two-dimensional slices.

We prove that at every infinite cardinality, every group which is neither abelian of exponent greater than two nor generalized dicyclic admits a graphical regular representation, settling the infinite-group part of Watkins's conjecture. We also determine the Cayley index of every infinite group: it is $1$, $2$, or $8$, according to its algebraic type, and in every case the index is attained by a connected Cayley graph. For every infinite group $G$ of cardinality $\kappa$, we construct $2^\kappa$ pairwise nonisomorphic Cayley graphs with exactly the unavoidable inverse-pair symmetries, diameter two, and $\kappa$ common neighbors at every distinct pair. The principal tool recovers a continuous ordinal hierarchy from an alternating adjacency baseline with bounded-degree errors: robust finite patterns identify the initial classes, successive twin quotients recover the layers, and their finite exception packets determine the translation action. The reconstruction applies without a group action and is stable under additional layerwise bounded-degree edits. Further results give closed Cantor-cube families with prescribed finite data in the regular cases, sharp cofinality-dependent graph properties, and optimal three-valued shortest-path metrics.

For a fixed digraph $F$, let $\operatorname{ex}_2^+(n,F)$ be the maximum of $\sum_{v\in V(D)}d_D^+(v)^2$ over all $n$-vertex $F$-free digraphs. Ai et al. [arXiv:2606.03520, 2026] asked for which self-converse tournament $F$ one can determine $\operatorname{ex}_2^+(n,F)$. Let $TT_r$ denote a transitive tournament on $r$ vertices and $\text{RT}_5$ denote a regular tournament on 5 vertices. Note that both $TT_r$ and $\text{RT}_5$ are self-converse. Iľkovič [arXiv:2609.05042, 2026] determined $\operatorname{ex}_2^+(n,F)$ for $F=TT_4$. In this paper, we determine $\operatorname{ex}_2^+(n,F)$ for $F$ being $TT_r$ or $\text{RT}_5$, and the latter confirms a conjecture due to Iľkovič.

We prove the real-rootedness conjecture of Lin, Ma, Ma, and Zhou (2021) for multiset Eulerian-Narayana polynomials, including simplicity and negativity of all zeros after removing the factor t. Using a forest decomposition and Lagrange inversion, we derive a differential formula and a first-order factorization of the differential operator. We use these factors to show that the nonconstant gamma-coefficients and ordered gamma-zeros are nondecreasing, with corresponding bounds given by the Narayana and Eulerian polynomials. The same factorization gives totally nonnegative coefficient transition matrices. We also prove strict interlacing of the zeros of generating polynomials arising from adjacent columns of the transition matrices.

We prove that every event invariant under swaps has probability zero or one for either the free or the wired spanning forest on any connected, locally finite, infinite network. Applications include the constancy of the number of components and of the total number of excessive ends. For the free forest, these results resolve longstanding questions of Benjamini, Lyons, Peres, and Schramm [Ann. Probab. (2001), Question 15.7] on constancy of the component count and of Lyons, Morris, and Schramm [Electron. J. Probab. (2008), Question 7.7] on the zero-one law for one-endedness. In the latter case, we obtain the stronger conclusion that the total number of excessive ends is deterministic.

Let $\delta^*_{k,r}$ be the fractional $K_r^{(k)}$-decomposition threshold in minimum codegree. For fixed $k\ge2$, $r\ge k+1$ and $\varepsilon>0$, we prove that every $n$-vertex $k$-uniform hypergraph $G$ with minimum codegree at least $(\delta^*_{k,r}+\varepsilon)n$ admits, with high probability, a fractional $K_r^{(k)}$-decomposition after retaining each edge independently with probability $p \ge C(\log n/n^{r-k})^{1/(\binom{r}{k}-1)}$. The bound on $p$ is sharp up to a constant factor, confirming the conjectured threshold order in the graph case. The proof develops a probabilistic matrix-scaling approach.

The discovery of Chair44 (Tsiokos, 2026) settled the three-dimensional einstein problem with a strongly aperiodic polyhedral monotile in $\mathbb{R}^3$. This note extends the underlying mechanism—the rep-$2^N$ chair $C_N = [0,2]^N \setminus (1,2]^N$ with corner/socket markings—to $\mathbb{R}^N$. Besides expository material (the rep-$2^N$ dissection and a conditional strong-aperiodicity theorem under lattice registration and hierarchical enforcement), the note makes a new computational contribution. We introduce a frame-marking formalism in which the marking of a tile is its full orientation frame and the matching rule is the contact language generated by the substitution itself; this makes the search for matching rules finite in every dimension. We give a finite certificate (coarsening closure, tightness, and a two-shell enclosure analysis) whose validity implies that every lattice-registered tiling by the marked tile is uniquely hierarchical, hence strongly aperiodic. For $N=3$ the certificate passes: it yields explicit facet matching rules on the 24 panels of $C_3$ (135 admissible facet-contact triples) and reproduces, from first principles and independently of published constructions, the Chair44 statistics 2388 $\to$ 44 admissible contacts (30 occurring), 33 one-shell clusters, 15 extendable, each forcing a unique supertile. Among the 2187 homochiral frame assignments of the 3D substitution with a translated central child, the certified one is unique up to conjugation. For $N=4$ the same pipeline is run on several structured families of frame assignments (canonical, $D_4$-, $Z_2\times Z_2$- and $Z_4$-symmetric, and a lift of the 3D solution); none is coarsening-closed, and we report the failure data. A self-similar marking of $C_4$ thus remains an explicitly finite, open computational problem, which we state precisely. Code: https://github.com/dimkadimon/Monotile-RN

In the fair dueling game, two players take turns shooting at each other, with the turn order defined by the following fairness criterion: the next shot is assigned to the player who is less likely to have won the duel so far. Cooper and Dutle showed that when both players have the same probability $p$ of a successful shot, the shooting sequence converges to the Thue–Morse sequence as $p \rightarrow 0$. We extend this model by forcing the sequence to begin with an arbitrary finite prefix before reverting to this fairness criterion. We prove that this initial disruption fundamentally alters the sequence's eventual behavior: in the $p \rightarrow 0$ limit, unless the initial prefix is itself a prefix of the Thue–Morse sequence or its complement, the resulting sequence is eventually periodic. Moreover, the least period is a power of 2 that is at most $4^{n+1}$, where $n$ is the length of the prefix, and the repeating block is itself a prefix of the Thue–Morse sequence.

Brandt [Discrete Math. 183 (1998) 17–25] conjectured that the least signless Laplacian eigenvalue of every regular triangle-free graph of order $n$ is at most $4n/25$. Using flag algebras, Balogh, Clemen, Lidický, Norin and Volec [SIAM J. Discrete Math. 37 (2023) 1173–1179] established the stronger bound $15n/94$ for all triangle-free graphs. We investigate the effect of additionally excluding pentagons and prove that every $\{C_3,C_5\}$-free graph $G$ of order $n$ satisfies $\qmin(G)<0.0569n$, without any regularity assumption. The proof combines seven-vertex flag inequalities with local Rayleigh constraints that retain the least eigenvalue throughout the counting argument. An exact integer certificate establishes the required inequality.

We systematically explore the relationship between edge-connectivity and Lin–Lu–Yau curvature for graphs. The intuition is that a connected graph with large Lin–Lu–Yau curvature should also have large edge-connectivity, and vice versa under suitable conditions. We prove that the edge-connectivity of a connected graph is bounded lower by the product of its minimum degree, its Lin–Lu–Yau curvature and the constant $\frac{3}{2}$. We also provide two lower bounds on the edge-connectivity of a graph that guarantee positive Lin–Lu–Yau curvature, and discuss the sharpness of these bounds.

For a positive integer $k$, let $\beta_k(G)$ be the maximum cardinality of a vertex set inducing maximum degree less than $k$, and let $SF_k(G)$ be the maximum cardinality of a minimal $k$-star-forming set. The known inequality $\beta_k(G)\le SF_k(G)$ suggests asking when equality holds. We place this question in the framework of upper domination: at $k=1$, $\beta_1(G)=\alpha(G)$ and $SF_1(G)=\Gamma(G)$, so the classical equality $\Gamma=\alpha$ on bipartite graphs is exactly the first member of the proposed hierarchy. We prove the equality for complete bipartite graphs for every $k$, obtaining \[ \beta_k(K_{a,b})=SF_k(K_{a,b})=\max\{a,b,2k-2\}\qquad(a,b\ge k), \] and record the elementary low-degree case $\Delta(G)<k$. For $k=2$ we derive certificate restrictions for minimal $2$-star-forming sets in bipartite graphs. For chain graphs we go further: we prove $\beta_2(G)=SF_2(G)$ and obtain an exact formula for their common value. The proof uses the nested-neighborhood structure together with a classification of witnesses to the indispensability of a high internal-degree vertex. We retain the equality problem for chain graphs as a conjecture only for $k\ge3$, and formulate the broader bipartite conjecture. We also determine the upper domination number of every rectangular grid and combine it with the known exact dissociation number to compare $\Gamma$, $\beta_2$, and $SF_2$. In particular, $\Gamma=\beta_2$ on every even-by-even rectangular grid, while $\beta_2\le SF_2$ always; this motivates a grid equality conjecture whose even-by-even case would yield a three-parameter identity.

Adapting the circle method developed by Frantzikinakis–Klurman–Moreira, we establish a sufficient criterion for a set of natural numbers to be multiplicatively recurrent. We apply this criterion to Piatetski–Shapiro sets with small exponents and to regular irrational nil-Bohr sets. We also study the value distribution of unimodular completely multiplicative functions along these sets.

Let $K_t$ be obtained by attaching a pendant path of length $t$ to a fixed rooted graph $K$. We prove that $\mathrm{Mo}(L(K_t))-\mathrm{Mo}(K_t)$, the difference between its line-graph and original Mostar indices, is exactly affine on each parity class beyond a sharp uniform cutoff. The slope depends on distance-level neighbor counts and is positive for every non-bipartite core. Triangle chains give infinitely many graphs with $\mathrm{Mo}(L(G))>\mathrm{Mo}(G)$ at every positive cyclomatic number $c$, with maximum degree three, resolving Alex-Indulal Problem 3.3. At zero slope, a cactus branch-mass formula decides equality; identical vertex profiles need not give identical intercepts. Fixed cores giving equality for all sufficiently long attachments exist exactly when $c$ is odd. Maximum degree three suffices, and at every even $c$ a binary-tree construction gives equality on one eventual parity class.

In this paper we define the ascending number and bridge number for virtual and welded spatial graphs. We prove that the former admits a lower bound in terms of the latter. This relationship allows us to compare different notions of unknotting virtual spatial graphs by crossing changes.

We study two Riordan arrays associated with the Motzkin-prefix sequence (\href{https://oeis.org/A005773}{OEIS A005773}). By connecting their production matrices through a simple boundary factorization, we obtain a period-three transformation on Motzkin endpoint distributions and exact formulas for paths by final height modulo~$3$. In addition, a unified factorization for the shifted Hankel matrices leads directly to explicit closed forms for the third and fourth Hankel determinants, $\Delta_n^{(3)}$ and~$\Delta_n^{(4)}$.

A $k$-arrangement of pseudolines is a set of bi-infinite curves in the plane such that any two of them intersect each other in exactly $k$ points, at which they cross, and it is simple if no three curves meet at a common point. Cyclic arrangements are the only simple $1$-arrangements of pseudolines that are unavoidable, in the Ramsey spirit: for each fixed $m\ge 1$, every sufficiently large simple $1$-arrangement of pseudolines has a cyclic subarrangement of size $m$. We show that, for every $m\ge 3$, the number of unavoidable simple $k$-arrangements of pseudolines of size $m$ grows exponentially with $k$, independently of $m$. For even $k$, we prove an analogous result for $k$-arrangements of pseudocircles.

For a graph G, the Boolean quadric polytope P(G) is the convex hull of the binary solutions of $y_{ij}=x_ix_j$ for $ij\in E(G)$, and Q(G) is its standard linear relaxation. For series-parallel graphs, Q(G) together with the odd-cycle inequalities describes P(G). Lee and Skipper showed vol(Q(G)) is polynomial-time computable for bounded treewidth and gave a closed formula for vol(P(G)) when G is a cycle. We resolve their question of giving an efficient algorithm for vol(P(G)) on series-parallel graphs. With $d=|V(G)|+|E(G)|$, we compute it with $O(d^7)$ arithmetic operations, and $O(d^5)$ if G is a cactus. The algorithm is a dynamic program over the series-parallel decomposition. The same framework computes vol(Q(G)) (hence the number of linear extensions of the incidence poset of G) with $O(d^4)$ operations, and $O(d^3)$ for cacti. With every $x_v$ fixed at 1/2, the recursion reduces to convolutions of univariate polynomials and computes the cut polytope volume of every series-parallel graph with m edges in $O(m^3)$ operations. We also study how much of Q(G) the polytope P(G) occupies. Short odd cycles matter more than long ones: in every graph, the odd-cycle inequalities of a cycle of length $\ell$ cut off at most a fraction $2^{\ell-1}/\ell!$ of Q(G). Hence vol(P(G))/vol(Q(G)) $\ge 1-\sum_C 2^{|C|-1}/|C|!$ for series-parallel G, with C ranging over its cycles. The ratio does not factor over cycles sharing a vertex: in a flower of $k$ copies of $C_\ell$ it decays like $\rho_\ell^k$ for an explicit rational $\rho_\ell$ below the ratio of $C_\ell$. Nevertheless, the small fractions cut off by many long cycles compound: in the worst case, triangle inequalities, or odd-cycle inequalities up to any fixed length, close no fixed fraction of the gap between Q(G) and P(G), and the volume ratio can be exponentially small in d.

Given a polygon in the plane with vertices on lattice points, Pick's theorem expresses its area A in terms of the number I of lattice points in its interior and the number B of lattice points in its boundary: A = I + B/2 - 1. We consider polygons in the plane whose edges are halfway between lattice points. For such a polygon that is also unimodular, we express its area A in terms of the number I of lattice points in its interior and the number V of its vertices: A = I - V/8 + 1/2 .

We study total path length in recursive trees with positive deterministic attachment weights. Writing $W_n=\sum_{i=1}^n w_i$ and $p_n=w_n/W_n$, we obtain exact martingale-innovation identities and a variance recurrence. Under the condition $p_n=O(n^{-1})$, centered total path length divided by $n$ converges almost surely and in $L^2$ to a nondegenerate random variable, and its variance is asymptotic to a positive constant times $n^2$. No polynomial asymptotic for $W_n$ is required. For power weights $w_i=i^\alpha$, the same argument applies to every real $\alpha$, including the critical and summable regimes beyond the positive-power cumulative-weight assumptions of existing profile theory. The expected average depth is logarithmic for $\alpha>-1$, iterated logarithmic for $\alpha=-1$, and bounded for $\alpha<-1$, while the global fluctuation scale remains linear throughout. In the summable regime we identify the random limit through the weighted depths of the infinite tree. The uniform case recovers the classical variance coefficient $2-\pi^2/6$. For linear weights we evaluate the coefficient as $8-2\pi^2/3$. Although this tree and a random binary search tree have identical insertion-depth marginals and expected total path length, their asymptotic variance coefficients differ by one. This gives an explicit comparison of global dependence that is invisible in individual depth distributions.

In this paper, we give exact combinatorial expressions for all CUE joint moments on the unit circle, in the case that all exponents are even positive integers, yielding the corresponding asymptotics. We offer two equivalent expressions for these quantities: the first is a sum involving Kostka numbers; the second is related to counting integer matrices (magic squares). As an application of our results, we unify many results in the literature, which were previously disconnected. Moreover, we prove a conjecture of Basor et al. (2018) regarding an irregular conformal block structure.

Let $X$ be a nodal curve, and $G$ be the graph dual to $X$. A stability condition on a graph $G$ is an assignment of integers to the biconnected subsets of vertices of $G$ satisfying some desired properties. Stability conditions yield degeneracy sets, which are certain collections of biconnected subsets of $G$, but degeneracy sets can also be described without a reference stability condition. We show that the degeneracy sets of a graph $G$ correspond to the single-element extensions of the graphic matroid $M(G)$. Some single-element extensions of the graphic matroid $M(G)$ can be oriented to be single-element extensions of the oriented graphic matroid $\mathcal{M}(G)$. Single-element extensions of oriented matroids are in bijection with subdivisions of the Lawrence polytope of the dual oriented matroid. We construct stability conditions on $G$ from subdivisions of the Lawrence polytope of the cographic matroid $\mathcal{M}^\ast(G)$. Finally, we show that any degeneracy set $\mathcal{D}$ of a graph $G$ that has the set of all biconnected subsets of $G$ as a lower bound in the poset of degeneracy sets corresponds to an orientable extension of $M(G)$.

Motivated by John Baez's question about seven exceptional coincidences among binomial coefficients, we construct seven exterior-power identities for representations of $A_5$. Their common symmetry comes from simplex edges and icosahedral vertices, axes, and faces, together with characters of stabilizers. We enumerate all ordered pairs of representation isomorphism classes in the prescribed dimensions. Apart from the pair of trivial actions, the permutation pair at $1540$ and the complex monomial pair at $7140$ are unique. At $3003$, Klein's twelve-point divisor gives a noncanonical intertwiner factored through polynomial division, differentiation, and residues.

We study negative edge correlation for two well-known models in statistical physics, the arboreal gas and the $q < 1$ random-cluster model. We show that after the edges $e$ and $f$ are removed, their Rayleigh difference is a crossing contribution minus the covariance of two endpoint-connectivity events. For the arboreal gas on an arbitrary finite graph, we analyze the two leading coefficients of this difference. The first is the classical transfer-current square. If it vanishes, the second has an electrical sum-of-squares formula. This proves negative edge correlation at sufficiently large fugacity whenever one of the two leading coefficients of the Rayleigh difference is nonzero, and characterizes simultaneous equality by an equipotential decomposition. For the random-cluster model on the complete graph $K_n$, we prove negative edge correlation throughout $q<1$ when all edge weights are at least $2$. In the uniform case, we prove something stronger:negative correlation holds when all the weights are equal and at least $1$. In particular, distinct edges in a connected spanning subgraph of $K_n$, weighted by a fixed fugacity, are negatively correlated for every $n$. We also explore what seems to be a general phenomenon: positive correlation of connectivity events. For the arboreal gas, we prove positive correlation for connectivity events at high fugacities. On lattices, the conjectured inequality would make the two-point function supermultiplicative and produce a convex inverse correlation length. Exact computations support the graph and matroid conjectures, including the Seymour–Welsh matroid $\mathcal S_8$, where ordinary edge-negative correlation fails. The results in this paper were derived by the authors without the use of Large Language models. The authors did benefit from using GPT-5 Pro for generating code to test out hypotheses as well as for simplifying the arguments.

For a graph $G=(V,E)$ of order $n$ and an integer $k$ between $1$ and $\lfloor\frac{n}{2}\rfloor$, its token graph $F_k(G)$ is the graph whose vertices consist of the $\binom{n}{k}$ $k$-subsets of $V$, and two vertices of $F_k(G)$ are adjacent whenever their symmetric difference is an edge in $E$. It was found that the algebraic connectivity of a graph is greater than or equal to that of its token graph, while the Laplacian spectral radius of a graph is less than or equal to that of its token graph. Moreover, a conjecture that the algebraic connectivity of a graph coincides with that of its token graph has been proved by using the theory of continuous Markov chains of random walks. In this paper, we derive some results about the algebraic connectivities of a graph and the same graph after adding new edges and their token graphs to obtain a combinatorial/algebraic proof. Besides, we provide some conditions under which the Laplacian spectral radius of a graph is less than that of its token graph. Finally, we characterize the graphs that have the same Laplacian spectral radius as their token graphs, including trees.

Børve, Horiatakis, and Kalck defined the magnitude of the module category of a representation-finite algebra as the sum of the entries of the inverse of the matrix of Hom dimensions between indecomposable modules. They showed it is an Euler characteristic, the alternating count of vertices, arrows, and meshes of the Auslander–Reiten quiver, and conjectured that it is at least the number of simple modules, with equality precisely for special biserial algebras. We prove this conjecture over an algebraically closed field. For a representation-directed algebra, we relate the magnitudes of the module category, the subcategory of modules without a given simple composition factor, and the ideal quotient by it. By the theory of hammocks of Ringel and Vossieck, this quotient is equivalent to a category of representations of a finite poset, whose magnitude we compute. The general case reduces to the representation-directed case through algebras obtained by grading a standard form.

We prove that every $n$-point set in the plane has a two-coloring with discrepancy $O(n^{1/4})$ for all triangles. This improves the previous bound $O(n^{1/4}\sqrt{\log n})$ and matches the classical lower bound.

We uncover a connection between the classical limit of the Magnus expansion in quantum field theory and the Malvenuto-Reutenauer Hopf algebra of permutations. We study a cubic scalar theory describing a massive particle coupled to massless scalar quanta, which captures the propagator structure relevant to classical scattering in gravity and scalar QED. Using a Schwinger parametrisation of the massive propagators, we compute the classical limit of Magnus amplitudes at tree level in this theory. First, we prove that all hyperclassical contributions cancel at the level of the Schwinger proper-time integrand, before integration. Then, we show that the classical Magnus amplitude precisely matches expectations from worldline quantum field theory. Both results rely on cancellations of disconnected diagrams which we relate to the Hopf algebra, specifically to the action of the adjoint first Eulerian projector, which annihilates the shuffle products associated with disconnected contributions. Moreover, the worldline diagrams include generic directed-tree topologies, whereas the field-theory diagrams only include directed chains. The correspondence we establish between the two therefore leads to a new formula for generic Murua coefficients in terms of simpler directed chain coefficients. Finally, we observe a suggestive connection between Magnus amplitudes and the web-mixing matrices governing the exponentiation of multiple Wilson lines.