This paper is motivated by two conjectures proposed by Bender et al. [‘Complemented zero-divisor graphs associated with finite commutative semigroups’, Comm. Algebra 52(7) (2024), 2852–2867], which have remained open questions. The first conjecture states that if the complemented zero-divisor graph $ G(S) $ of a commutative semigroup $ S $ with a zero element has clique number three or greater, then the reduced graph $ G_r(S) $ is isomorphic to the graph $ G(\mathcal {P}(n)) $. The second conjecture asserts that if $ G(S) $ is a complemented zero-divisor graph with clique number three or greater, then $ G(S) $ is uniquely complemented. We construct a commutative semigroup $ S $ with a zero element that serves as a counter-example to both conjectures.

We establish several new results on the existence of probability distributions on the independent sets in triangle-free graphs where each vertex is present with a given probability. This setting was introduced and studied under the name of “fractional coloring with local demands” by Kelly and Postle and is closely related to the well-studied fractional chromatic number of graphs.

Our first main result strengthens Shearer’s classic bound on independence number, proving that for every triangle-free graph G there exists a distribution over independent sets where each vertex v appears with probability $(1-o(1))\frac {\ln d_G(v)}{d_G(v)}$, resolving a conjecture by Kelly and Postle. This in turn implies new upper bounds on the fractional chromatic number of triangle-free graphs with a prescribed number of vertices or edges, which resolves a conjecture by Cames van Batenburg et al. and addresses yet another one by the same authors.

Our second main result resolves Harris’ conjecture on triangle-free d-degenerate graphs, showing that such graphs have fractional chromatic number at most $(4+o(1))\frac {d}{\ln d}$. As previously observed by various authors, this in turn has several interesting consequences. A notable example is that every triangle-free graph with minimum degree d contains a bipartite induced subgraph of minimum degree $\Omega (\log d)$. This settles a conjecture by Esperet, Kang, and Thomassé.

The main technique employed to obtain our results is the analysis of carefully designed random processes on vertex-weighted triangle-free graphs that preserve weights in expectation. The analysis of these processes yields weighted generalizations of the aforementioned results that may be of independent interest.

For a family $\mathcal {F}$ of graphs, let ${\mathrm {ex}}(n,\mathcal {F})$ denote the maximum number of edges in an n-vertex graph which contains none of the members of $\mathcal {F}$ as a subgraph. A longstanding problem in extremal graph theory asks to determine the function ${\mathrm {ex}}(n,\{C_3,C_4\})$. Here we give a new construction for dense graphs of girth at least five with arbitrary number of vertices, providing the first improvement on the lower bound of ${\mathrm {ex}}(n,\{C_3,C_4\})$ since 1976. As a corollary, this yields a negative answer to a problem in Chung-Graham [3].

The hard-core model has as its configurations the independent sets of some graph instance $G$. The probability distribution on independent sets is controlled by a ‘fugacity’ $\lambda \gt 0$, with higher $\lambda$ leading to denser configurations. We investigate the mixing time of Glauber (single-site) dynamics for the hard-core model on restricted classes of bounded-degree graphs in which a particular graph $H$ is excluded as an induced subgraph. If $H$ is a subdivided claw then, for all $\lambda$, the mixing time is $O(n\log n)$, where $n$ is the order of $G$. This extends a result of Chen and Gu for claw-free graphs. When $H$ is a path, the set of possible instances is finite. For all other $H$, the mixing time is exponential in $n$ for sufficiently large $\lambda$, depending on $H$ and the maximum degree of $G$.

Normal matrices, or matrices which commute with their adjoints, are of fundamental importance in pure and applied mathematics. In this paper, we study a natural functional on the space of square complex matrices whose global minimizers are normal matrices. We show that this functional, which we refer to as the non-normal energy, has incredibly well-behaved gradient descent dynamics: despite it being nonconvex, we show that the only critical points of the non-normal energy are the normal matrices, and that its gradient descent trajectories fix matrix spectra and preserve the subset of real matrices. We also show that, even when restricted to the subset of unit Frobenius norm matrices, the gradient flow of the non-normal energy retains many of these useful properties. This is applied to prove that low-dimensional homotopy groups of spaces of unit norm normal matrices vanish; for example, we show that the space of $d \times d$ complex unit norm normal matrices is simply connected for all $d \geq 2$. Finally, we consider the related problem of balancing a weighted directed graph – that is, readjusting its edge weights so that the weighted in-degree and out-degree are the same at each node. We adapt the non-normal energy to define another natural functional whose global minima are balanced graphs and show that gradient descent of this functional always converges to a balanced graph, while preserving graph spectra and realness of the weights. Our results were inspired by concepts from symplectic geometry and Geometric Invariant Theory, but we mostly avoid invoking this machinery and our proofs are generally self-contained.

We prove a new tableaux formula for the symmetric Macdonald polynomials $P_{\lambda }(X;q,t)$ that has considerably fewer terms and simpler weights than previously existing formulas. Our formula is a sum over certain sorted non-attacking tableaux, weighted by the queue inversion statistic $\operatorname {\mathrm {\texttt {quinv}}}$. The $\operatorname {\mathrm {\texttt {quinv}}}$ statistic originates from a formula for the modified Macdonald polynomials $\widetilde {H}_{\lambda }(X;q,t)$ due to Ayyer, Martin, and the author (2022), and is naturally related to the dynamics of the asymmetric simple exclusion process (ASEP) on a circle.

We prove our results by introducing probabilistic operators that act on non-attacking tableaux to generate a set of tableaux whose weighted sum equals $P_{\lambda }(X;q,t)$. These operators are a modification of the inversion flip operators of Loehr and Niese (2012), which yield an involution on tableaux that preserves the major index statistic but fails to preserve the non-attacking condition. Our tableaux are in bijection with the multiline queues introduced by Martin (2020), allowing us to derive an alternative multiline queue formula for $P_{\lambda }(X;q,t)$. Finally, our formula recovers an alternative formula for the Jack polynomials $J_{\lambda }(X;\alpha )$ due to Knop and Sahi (1996) using the same queue inversion statistic.

For each set X, an X-split is a partition of X into two parts. For each X-split S and each subset $Y\subseteq X$, the restriction of S on Y is the Y-split whose parts are obtained by intersecting the parts of S with Y. For a graph G with vertex set V, the G-coboundary size of a V-split S is the number of edges in G having non-empty intersections with both parts of S. Let T be a tree without degree-two vertices, and let V and L denote its vertex set and leaf set, respectively. For each positive integer k, a k-split on T is an L-split that is the restriction of a V-split with T-coboundary size k, while a score-k split on T is a k-split on T that is not any k′-split for any integer $k' \lt k$. Buneman’s split equivalence theorem states that the tree T is entirely encoded by its system of score-1 splits. We identify the unique exceptional case in which the tree T is not determined by its score-2 split system. To explore how our work can be extended to more general tree isomorphism problems, we propose several conjectures and open problems related to set systems and generalized Buneman graphs.

Negami found an elegant splitting formula for the Tutte polynomial. We present an analogue of this for Bollobás and Riordan’s ribbon graph polynomial, and for the transition polynomial. From this we deduce a splitting formula for the Jones polynomial.

A simple k-coloring of a multigraph G is a decomposition of the edge multiset as a disjoint sum of k simple graphs which are referred to as colors. A subgraph H of a multigraph G is called multicolored if its edges receive distinct colors in a given simple k-coloring of G. In 2004, Keevash–Saks–Sudakov–Verstraëte introduced the k-color Turán number ${\text {ex}}_k(n,H)$, which denotes the maximum number of edges in an n-vertex multigraph that has a simple k-coloring containing no multicolored copies of H. They made a conjecture for any $r\geq 3$ and r-color-critical graph $H,$ that in the range of $k\geq \frac {r-1}{r-2}(e(H)-1)$, if n is sufficiently large, then ${\text {ex}}_k(n, H)$ is achieved by the multigraph consisting of k colors all of which are identical copies of the Turán graph $T_{r-1}(n)$. In this article, we show that this holds in the range of $k\geq 2\frac {r-1}{r}(e(H)-1)$, significantly improving earlier results. Our proof combines the stability argument of Chakraborti–Kim–Lee–Liu–Seo with a novel graph packing technique for embedding multigraphs.

We derive large- and moderate-deviation results in random networks given as planar directed navigations on homogeneous Poisson point processes. In this non-Markovian routing scheme, starting from the origin, at each consecutive step a Poisson point is joined by an edge to its nearest Poisson point to the right within a cone. We establish precise exponential rates of decay for the probability that the vertical displacement of the random path is unexpectedly large. The proofs rest on controlling the dependencies of the individual steps and the randomness in the horizontal displacement as well as renewal-process arguments.

We consider the random series–parallel graph introduced by Hambly and Jordan (2004 Adv. Appl. Probab. 36, 824–838), which is a hierarchical graph with a parameter $p\in [0, \, 1]$. The graph is built recursively: at each step, every edge in the graph is either replaced with probability p by a series of two edges, or with probability $1-p$ by two parallel edges, and the replacements are independent of each other and of everything up to then. At the nth step of the recursive procedure, the distance between the extremal points on the graph is denoted by $D_n (p)$. It is known that $D_n(p)$ possesses a phase transition at $p=p_c \;:\!=\;\frac{1}{2}$; more precisely, $\frac{1}{n}\log {{\mathbb{E}}}[D_n(p)] \to \alpha(p)$ when $n \to \infty$, with $\alpha(p) >0$ for $p>p_c$ and $\alpha(p)=0$ for $p\le p_c$. We study the exponent $\alpha(p)$ in the slightly supercritical regime $p=p_c+\varepsilon$. Our main result says that as $\varepsilon\to 0^+$, $\alpha(p_c+\varepsilon)$ behaves like $\sqrt{\zeta(2) \, \varepsilon}$, where $\zeta(2) \;:\!=\; \frac{\pi^2}{6}$.

Asymptotic dimension and Assouad–Nagata dimension are measures of the large-scale shape of a class of graphs. Bonamy, Bousquet, Esperet, Groenland, Liu, Pirot, and Scott [J. Eur. Math. Society] showed that any proper minor-closed class has asymptotic dimension 2, dropping to 1 only if the treewidth is bounded. We improve this result by showing it also holds for the stricter Assouad–Nagata dimension. We also characterise when subdivision-closed classes of graphs have bounded Assouad–Nagata dimension.

We prove a ‘Whitney’ presentation, and a ‘Coulomb branch’ presentation, for the torus equivariant quantum K theory of the Grassmann manifold $\mathrm {Gr}(k;n)$, inspired from physics, and stated in an earlier paper. The first presentation is obtained by quantum deforming the product of the Hirzebruch $\lambda _y$ classes of the tautological bundles. In physics, the $\lambda _y$ classes arise as certain Wilson line operators. The second presentation is obtained from the Coulomb branch equations involving the partial derivatives of a twisted superpotential from supersymmetric gauge theory. This is closest to a presentation obtained by Gorbounov and Korff, utilizing integrable systems techniques. Algebraically, we relate the Coulomb and Whitney presentations utilizing transition matrices from the (equivariant) Grothendieck polynomials to the (equivariant) complete homogeneous symmetric polynomials. Along the way, we calculate K-theoretic Gromov-Witten invariants of wedge powers of the tautological bundles on $\mathrm {Gr}(k;n)$, using the ‘quantum=classical’ statement.

Given two graphs G and H, the Ramsey number $R(G,H)$ is the smallest positive integer N such that every graph of order N contains G or its complement contains H as a subgraph. Let $C_n$ denote the cycle on n vertices and let $tW_{2m+1}$ denote the disjoint union of t copies of the $(2m+2)$-vertex wheel $W_{2m+1}$. We show that for integers $m\ge 1$, $t\ge 2$ and $n\ge (6m+3)t-6m+999$,

$$ \begin{align*} R(C_n, tW_{2m+1})=3n+t-3. \end{align*} $$

This result extends several previous results and settles a conjecture posed by Sudarsana [‘A note on the Ramsey number for cycle with respect to multiple copies of wheels’, Electron. J. Graph Theory Appl. 9(2) (2021), 561–566].

We give combinatorially controlled series solutions to Dyson–Schwinger equations with multiple insertion places using tubings of rooted trees and investigate the algebraic relation between such solutions and the renormalization group equation.

We show that for every non-spherical set X in $\mathbb {E}^d$, there exists a natural number m and a red/blue-coloring of $\mathbb {E}^n$ for every n such that there is no red copy of X and no blue progression of length m with each consecutive point at distance $1$. This verifies a conjecture of Wu and the first author.

Asymptotic properties of random graph sequences, like the occurrence of a giant component or full connectivity in Erdös–Rényi graphs, are usually derived with very specific choices for the defining parameters. The question arises as to what extent those parameter choices may be perturbed without losing the asymptotic property. For two sequences of graph distributions, asymptotic equivalence (convergence in total variation) and contiguity have been considered by Janson (2010) and others; here we use so-called remote contiguity to show that connectivity properties are preserved in more heavily perturbed Erdös–Rényi graphs. The techniques we demonstrate here with random graphs also extend to general asymptotic properties, e.g. in more complex large-graph limits, scaling limits, large-sample limits, etc.

La façon la plus simple de faire d’un graphe fini connexe G un système dynamique est de lui donner une polarisation, c’est-à-dire un ordre cyclique des arêtes incidentes à chaque sommet. L’espace de phase $\mathcal {P}(G)$ d’un graphe consiste en toutes les paires $(v,e)$ où v est un sommet et e une arête incidente à v. Elle donne donc la position et le vecteur initiaux. Une telle condition est équivalente à une arête que l’on munit d’une orientation $e_{\mathcal O}$. Avec la polarisation, chaque donnée initiale mène à une marche à gauche en tournant à gauche à chaque sommet rencontré, ou en rebondissant s’il n’y a en ce sommet aucune autre arête. Une marche à gauche est appelée complète si elle couvre toutes les arêtes de G (pas nécessairement dans les deux sens). Nous définissons la valence d’un sommet comme le nombre d’arêtes adjacentes à ce sommet, et la valence d’un graphe comme étant la moyenne des valences de ses sommets. Dans cet article, nous démontrons que si un graphe plongé dans une surface orientée fermée de genre g possède une marche à gauche complète, alors sa valence est d’au plus $1 + \sqrt {6g+1}$. Nous prouvons de plus que ce résultat est optimal pour une infinité de genres g et qu’il est asymptotiquement optimal lorsque $g \to + \infty $. Cela mène à des obstructions pour les plongements de graphes sur une surface. Puisque vérifier si un graphe polarisé possède ou non une marche à gauche complète s’opère en temps au plus $4N$, où N est le nombre d’arêtes (il suffit de le vérifier sur les deux orientations d’une seule arête donnée), cette obstruction est particulièrement efficace. Ce problème trouve sa motivation dans ses conséquences intéressantes sur ce que nous appellerons ici l’ergodicité topologique d’un système conservatif, par exemple un système hamiltonien H en dimension deux où l’existence d’une marche complète à gauche correspond à une orbite du système topologiquement ergodique, donc une orbite qui visite toute la topologie de la surface. Nous nous limitons ici à la dimension $2$, mais une généralisation de cette théorie devrait tenir pour des systèmes hamiltoniens autonomes sur une variété symplectique de dimension arbitraire.

Minimal kinematics identifies likelihood degenerations where the critical points are given by rational formulas. These rest on the Horn uniformization of Kapranov–Huh. We characterize all choices of minimal kinematics on the moduli space $\mathcal{M}_{0,n}$. These choices are motivated by the CHY model in physics and they are represented combinatorially by 2-trees. We compute 2-tree amplitudes, and we explore extensions to non-planar on-shell diagrams, here identified with the hypertrees of Castravet–Tevelev.

The Erdős-Sós Conjecture states that every graph with average degree exceeding $k-1$ contains every tree with $k$ edges as a subgraph. We prove that there are $\delta \gt 0$ and $k_0\in \mathbb N$ such that the conjecture holds for every tree $T$ with $k \ge k_0$ edges and every graph $G$ with $|V(G)| \le (1+\delta )|V(T)|$.