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Spectra and eigenspaces from regular partitions of Cayley (di)graphs of permutation groupsJun 13 2019In this paper, we present a method to obtain regular (or equitable) partitions of Cayley (di)graphs (that is, graphs, digraphs, or mixed graphs) of permutation groups on $n$ letters. We prove that every partition of the number $n$ gives rise to a regular ... More

Multivariate polynomials for generalized permutohedraJun 13 2019Using the notion of Mahonian statistic on acyclic posets, we introduce a $q$-analogue of the $h$-polynomial of a simple generalized permutohedron. We focus primarily on the case of nestohedra and on explicit computations for many interesting examples, ... More

Concentration estimates for algebraic intersectionsJun 13 2019We present an approach over arbitrary fields to bound the degree of intersection of families of varieties in terms of how these concentrate on algebraic sets of smaller codimension.

Fractional Local DimensionJun 13 2019The original notion of dimension for posets was introduced by Dushnik and Miller in 1941 and has been studied extensively in the literature. In 1992, Brightwell and Scheinerman developed the notion of fractional dimension as the natural linear programming ... More

On co-minimal pairs in abelian groupsJun 13 2019A pair of non-empty subsets $(W,W')$ in an abelian group $G$ is a complement pair if $W+W'=G$. $W'$ is said to be minimal to $W$ if $W+(W'\setminus \{w'\}) \neq G, \forall \,w'\in W'$. In general, given an arbitrary subset in a group, the existence of ... More

On the Walks and Bipartite Double Coverings of Graphs with the same Main EigenspaceJun 13 2019The main eigenvalues of a graph $G$ are those eigenvalues of the $(0,1)$-adjacency matrix $\mathbf A$ having a corresponding eigenvector not orthogonal to $\mathbf j = (1,\dots,1)$. The CDC of a graph $G$ is the direct product $G\times K_2$. The main ... More

Characteristic Power Series of Graph LimitsJun 13 2019In this note, we show how to obtain a ``characteristic power series'' of graphons -- infinite limits of graphs -- as the limit of normalized reciprocal characteristic polynomials. This leads to a characterization of graph quasi-randomness and another ... More

Hypersimplicial subdivisionsJun 13 2019Let $\pi:{\mathbb R}^n \to {\mathbb R}^d$ be any linear projection, let $A$ be the image of the standard basis. Motivated by Postnikov's study of postitive Grassmannians via plabic graphs and Galashin's connection of plabic graphs to slices of zonotopal ... More

The rank of sparse random matricesJun 13 2019Generalising prior work on the rank of random matrices over finite fields [Coja-Oghlan and Gao 2018], we determine the rank of a random matrix with prescribed numbers of non-zero entries in each row and column over any field. The rank formula turns out ... More

Graphs of bounded depth-$2$ rank-brittlenessJun 13 2019We characterize classes of graphs closed under taking vertex-minors and having no $P_n$ and no disjoint union of $n$ copies of the $1$-subdivision of $K_{1,n}$ for some $n$. Our characterization is described in terms of a tree of radius $2$ whose leaves ... More

Smooth digraphs modulo primitive positive constructabilityJun 13 2019We consider the poset that arises from ordering finite smooth digraphs via pp-constructability. We give a complete description of this poset and, in particular, we prove that it is a distributive lattice. Moreover, we show that in order to separate two ... More

Post-Processing of High-Dimensional DataJun 13 2019Scientific computations or measurements may result in huge volumes of data. Often these can be thought of representing a real-valued function on a high-dimensional domain, and can be conceptually arranged in the format of a tensor of high degree in some ... More

A Semi-strong Perfect Digraph TheoremJun 13 2019Reed showed that, if two graphs are $P_4$-isomorphic, then either both are perfect or none of them is. In this note we will derive an analogous result for perfect digraphs.

Nearly all cacti are edge intersection hypergraphs of 3-uniform hypergraphsJun 13 2019If ${\cal H}=(V,{\cal E})$ is a hypergraph, its edge intersection hypergraph $EI({\cal H})=(V,{\cal E}^{EI})$ has the edge set ${\cal E}^{EI}=\{e_1 \cap e_2 \ |\ e_1, e_2 \in {\cal E} \ \wedge \ e_1 \neq e_2 \ \wedge \ |e_1 \cap e_2 |\geq2\}$. Using the ... More

On the 4-color theorem for signed graphsJun 13 2019There are several ways to generalize graph coloring to signed graphs. M\'a\v{c}ajov\'a, Raspaud and \v{S}koviera introduced one of them and conjectured that in this setting, for signed planar graphs four colors are always enough, generalising thereby ... More

An Asymmetric Random Rado Theorem: 1-statementJun 13 2019A classical result by Rado characterises the so-called partition-regular matrices $A$, i.e.\ those matrices $A$ for which any finite colouring of the positive integers yields a monochromatic solution to the equation $Ax=0$. We study the {\sl asymmetric} ... More

Vertex properties of maximum scattered linear sets of $\mathrm{PG}(1,q^n)$Jun 13 2019In this paper we investigate the geometric properties of the configuration consisting of a $k$-subspace $\Gamma$ and a canonical subgeometry $\Sigma$ in $\mathrm{PG}(n-1,q^n)$, with $\Gamma\cap\Sigma=\emptyset$. The idea motivating is that such properties ... More

On Convex Graphs Having Plane Spanning Subgraph of Certain TypeJun 13 2019Motivated by a result of [17], we determine necessary and sufficient conditions on $F\/$ with $|E(F)| \leq n-1\/$ for which $K_n - F\/$ admits a $g$-angulation. For $|E(F)| \geq n\/$, we investigate the possibility of placing $F\/$ in $K_n\/$ such that ... More

On Edge-Partitioning of Complete Geometric Graphs into Plane TreesJun 13 2019In response to a well-known open question ``Does every complete geometric graph on $2n\/$ vertices have a partition of its edge set into $n\/$ plane spanning trees?" we provide an affirmative answer when the complete geometry graph is in the regular wheel ... More

On the first fall degree of summation polynomialsJun 13 2019We improve on the first fall degree bound of polynomial systems that arise from a Weil descent along Semaev's summation polynomials relevant to the solution of the Elliptic Curve Discrete Logarithm Problem via Gr\"obner basis algorithms.

Counting integer points of flow polytopesJun 13 2019The Baldoni--Vergne volume and Ehrhart polynomial formulas for flow polytopes are significant in at least two ways. On one hand, these formulas are in terms of Kostant partition functions, connecting flow polytopes to this classical vector partition function ... More

On discrete idempotent pathsJun 13 2019The set of discrete lattice paths from (0, 0) to (n, n) with North and East steps (i.e. words w $\in$ { x, y } * such that |w| x = |w| y = n) has a canonical monoid structure inherited from the bijection with the set of join-continuous maps from the chain ... More

Hypercontractivity for global functions and sharp thresholdsJun 13 2019The classical hypercontractive inequality for the noise operator on the discrete cube plays a crucial role in many of the fundamental results in the Analysis of Boolean functions, such as the KKL (Kahn-Kalai-Linial) theorem, Friedgut's junta theorem and ... More

Signed Hultman Numbers and Signed Generalized Commuting Probability in Finite GroupsJun 13 2019Let G be a finite group. Let pi be a permutation from S{n}. We study the distribution of probabilities of equality a{1} a{2} ...a{n-1}a{n}=a{pi{1}}^{epsilon{1}} a{pi_{2}}^{epsilon{2}}...a{pi{n-1}}^{epsilon_{n-1}} a_{pi_{n}}^{epsilon{n}}, when pi varies ... More

Binomial edge ideals of cographsJun 13 2019We determine the Castelnuovo--Mumford regularity of binomial edge ideals of complement reducible graphs (cographs). On $n$ vertices the maximum regularity is essentially $2n/3$. Independently of the number of vertices, we also bound the regularity by ... More

Fractional cocoloring of graphsJun 13 2019The cochromatic number $Z(G)$ of a graph $G$ is the fewest number of colors needed to color the vertices of $G$ so that each color class is a clique or an independent set. In a fractional cocoloring of $G$ a non-negative weight is assigned to each clique ... More

Combinatorially equivalent hyperplane arrangementsJun 13 2019We study the combinatorics of hyperplane arrangements over arbitrary fields. Specifically, we determine in which situation an arrangement and its reduction modulo a prime number have isomorphic lattices via the use of minimal strong $\sigma$-Gr\"obner ... More

Fixed-Parameter Tractability of Graph Deletion Problems over Data StreamsJun 13 2019In this work, we initiate a systematic study of parameterized streaming complexity of graph deletion problems: ${\cal F}$-Subgraph Deletion, ${\cal F}$-Minor Deletion and Cluster Vertex Deletion in the four most well-studied streaming models: the EA (edge ... More

Brouwer's conjecture holds asymptotically almost surelyJun 12 2019We show that for a sequence of random graphs Brouwer's conjecture holds true with probability tending to one as the number of vertices tends to infinity. Surprisingly, it was found that a similar statement holds true for weighted graphs with possible ... More

Multicolor Ramsey numbers of cycles in Gallai coloringsJun 12 2019For a graph $H$ and an integer $k\ge1$, the $k$-color Ramsey number $R_k(H)$ is the least integer $N$ such that every $k$-coloring of the edges of the complete graph $K_N$ contains a monochromatic copy of $H$. Let $C_m$ denote the cycle on $m\ge4 $ vertices. ... More

Higher extensions for gentle algebrasJun 12 2019In this paper we determine extensions of higher degree between indecomposable modules over gentle algebras. In particular, our results show how such extensions either eventually vanish or become periodic. We give a geometric interpretation of vanishing ... More

Voronoi conjecture for five-dimensional parallelohedraJun 12 2019We prove the Voronoi conjecture for five-dimensional parallelohedra. Namely, we show that if a convex five-dimensional polytope $P$ tiles $\mathbb{R}^5$ with translations, then $P$ is an affine image of the Dirichlet-Voronoi cell for a five-dimensional ... More

On the joint distribution of cyclic valleys and excedances over conjugacy classes of $\mathfrak{S}_{n}$Jun 12 2019We derive a formula expressing the joint distribution of the cyclic valley number and excedance number statistics over a fixed conjugacy class of the symmetric group in terms of Eulerian polynomials. Our proof uses a slight extension of Sun and Wang's ... More

Global optimization using Sobol indicesJun 12 2019We propose and assess a new global (derivative-free) optimization algorithm, inspired by the LIPO algorithm, which uses variance-based sensitivity analysis (Sobol indices) to reduce the number of calls to the objective function. This method should be ... More

Small-Support Uncertainty Principles on $\mathbb{Z}/p$ over Finite FieldsJun 12 2019We establish an uncertainty principle for functions $f: \mathbb{Z}/p \rightarrow \mathbb{F}_q$ with constant support (where $p \mid q-1$). In particular, we show that for any constant $S > 0$, functions $f: \mathbb{Z}/p \rightarrow \mathbb{F}_q$ for which ... More

Sharp thresholds for nonlinear Hamiltonian cycles in hypergraphsJun 12 2019For positive integers $r > \ell$, an $r$-uniform hypergraph is called an $\ell$-cycle if there exists a cyclic ordering of its vertices such that each of its edges consists of $r$ consecutive vertices, and such that every pair of consecutive edges (in ... More

Next-to$^k$ leading log expansions by chord diagramsJun 12 2019Green functions in a quantum field theory can be expanded as bivariate series in the coupling and a scale parameter. The leading logs are given by the main diagonal of this expansion, i.e. the subseries where the coupling and the scale parameter appear ... More

Biased random k-SATJun 12 2019The basic random $k$-SAT problem is: Given a set of $n$ Boolean variables, and $m$ clauses of size $k$ picked uniformly at random from the set of all such clauses on our variables, is the conjunction of these clauses satisfiable? Here we consider a variation ... More

On the Universal Near-Shortest Simple Paths ProblemJun 12 2019This article generalizes the Near-Shortest Paths Problem introduced by Byers and Waterman in 1984 using concepts of the Universal Shortest Path Problem established by Turner and Hamacher in 2011. The generalization covers a variety of shortest path problems ... More

Odd cycles in subgraphs of sparse pseudorandom graphsJun 12 2019We answer two extremal questions about odd cycles that naturally arise in the study of sparse pseudorandom graphs. Let $\Gamma$ be an $(n,d,\lambda)$-graph, i.e., $n$-vertex, $d$-regular graphs with all nontrivial eigenvalues in the interval $[-\lambda,\lambda]$. ... More

The complexity of the vertex-minor problemJun 12 2019A graph H is a vertex-minor of a graph G if it can be reached from G by the successive application of local complementations and vertex deletions. Vertex-minors have been the subject of intense study in graph theory over the last decades and have found ... More

Broadcasts on Paths and CyclesJun 12 2019A broadcast on a graph $G=(V,E)$ is a function $f: V\longrightarrow \{0,\ldots,\operatorname{diam}(G)\}$ such that $f(v)\leq e_G(v)$ for every vertex $v\in V$, where$\operatorname{diam}(G)$ denotes the diameter of $G$ and $e_G(v)$ the eccentricity of ... More

Fault-Tolerant Path-Embedding of Twisted Hypercube-Like Networks THLNsJun 12 2019The twisted hypercube-like networks($THLNs$) contain several important hypercube variants. This paper is concerned with the fault-tolerant path-embedding of $n$-dimensional($n$-$D$) $THLNs$. Let $G_n$ be an $n$-$D$ $THLN$ and $F$ be a subset of $V(G_n)\cup ... More

On the WalkerMaker-WalkerBreaker gamesJun 12 2019We study the unbiased WalkerMaker-WalkerBreaker games on the edge set of the complete graph on $n$ vertices, $K_n$, a variant of well-known Maker-Breaker positional games, where both players have the restriction on the way of playing. Namely, each player ... More

Approximating the Orthogonality Dimension of Graphs and HypergraphsJun 12 2019A $t$-dimensional orthogonal representation of a hypergraph is an assignment of nonzero vectors in $\mathbb{R}^t$ to its vertices, such that every hyperedge contains two vertices whose vectors are orthogonal. The orthogonality dimension of a hypergraph ... More

Nearly Finitary MatroidsJun 12 2019In this thesis, we study nearly finitary matroids by introducing new definitions and prove various properties of nearly finitary matroids. In 2010, an axiom system for infinite matroids was proposed by Bruhn et al. We use this axiom system for this thesis. ... More

Communication-Efficient Accurate Statistical EstimationJun 12 2019When the data are stored in a distributed manner, direct application of traditional statistical inference procedures is often prohibitive due to communication cost and privacy concerns. This paper develops and investigates two Communication-Efficient ... More

Homological Connectivity in Random Čech ComplexesJun 11 2019Jun 13 2019We study the homology of random \v{C}ech complexes generated by a homogeneous Poisson process. We focus on 'homological connectivity' - the stage where the random complex is dense enough, so that its homology "stabilizes" and becomes isomorphic to that ... More

Homological Connectivity in Čech ComplexesJun 11 2019We study the homology of random \v{C}ech complexes generated by a homogeneous Poisson process. We focus on 'homological connectivity' - the stage where the random complex is dense enough, so that its homology "stabilizes" and becomes isomorphic to that ... More

A note on extensions of multilinear maps defined on multilinear varietiesJun 11 2019Let $G_1, \dots, G_k$ be finite-dimensional vector spaces over a finite field $\mathbb{F}$. A multilinear variety of codimension $d$ is a subset of $G_1 \times \dots \times G_k$ defined as the zero set of $d$ forms, each of which is multilinear on some ... More

The Kőnig Graph ProcessJun 11 2019Say that a graph G has property $\mathcal{K}$ if the size of its maximum matching is equal to the order of a minimal vertex cover. We study the following process. Set $N:= \binom{n}{2}$ and let $e_1, e_2, \dots e_{N}$ be a uniformly random ordering of ... More

Enumerating linear systems on graphsJun 11 2019The divisor theory of graphs views a finite connected graph $G$ as a discrete version of a Riemann surface. Divisors on $G$ are formal integral combinations of the vertices of $G$, and linear equivalence of divisors is determined by the discrete Laplacian ... More

The $h^*$-polynomials of locally anti-blocking lattice polytopes and their $γ$-positivityJun 11 2019A lattice polytope $\mathcal{P} \subset \mathbb{R}^d$ is called a locally anti-blocking polytope if for any closed orthant $\mathbb{R}^d_{\varepsilon}$ in $\mathbb{R}^d$, $\mathcal{P} \cap \mathbb{R}^d_{\varepsilon}$ is unimodularly equivalent to an anti-blocking ... More

Areas of triangles and SL_2 actions in finite ringsJun 11 2019In Euclidean space, one can use the dot product to give a formula for the area of a triangle in terms of the coordinates of each vertex. Since this formula involves only addition, subtraction, and multiplication, it can be used as a definition of area ... More

Matricial characterization of tournaments with maximum number of diamondsJun 11 2019A diamond is a $4$-tournament which consists of a vertex dominating or dominated by a $3$-cycle. Assuming the existence of skew-conference matrices, we give a complete characterization of $n$-tournaments with the maximum number of diamonds when $n\equiv0\pmod{4}$ ... More

Distance Matrix of a Class of Completely Positive Graphs: Determinant and InverseJun 11 2019A real symmetric matrix $A$ is said to be completely positive if it can be written as $BB^t$ for some (not necessarily square) nonnegative matrix $B$. A simple graph $G$ is called a completely positive graph if every doubly nonnegative matrix realization ... More

An explicit characterization of arc-transitive circulantsJun 11 2019A reductive characterization of arc-transitive circulants was given independently by Kovacs in 2004 and the first author in 2005. In this paper, we give an explicit characterization of arc-transitive circulants and their automorphism groups. Based on ... More

A Graph-theoretic Method to Define any Boolean Operation on PartitionsJun 11 2019The lattice operations of join and meet were defined for set partitions in the nineteenth century, but no new logical operations on partitions were defined and studied during the twentieth century. Yet there is a simple and natural graph-theoretic method ... More

Translation hyperovals and $\mathbb{F}_2$-linear sets of pseudoregulus typeJun 11 2019In this paper, we study translation hyperovals in PG$(2,q^k)$. The main result of this paper characterises the point sets defined by translation hyperovals in the Andr\'e/Bruck-Bose representation. We show that the affine point sets of translation hyperovals ... More

Identities involving Schur functions and their applications to a shuffling theoremJun 11 2019Lai and Rohatgi found a shuffling theorem, which generalizes earlier work of Ciucu on an enumeration of lozenge tilings of a hexagon with a fern removed from a center. They presented many generalizations of the shuffling theorem and also suggested a conjecture ... More

Three product formulas for ratios of tiling counts of hexagons with collinear holesJun 11 2019Rosengren found an explicit formula for a certain weighted enumeration of lozenge tilings of a hexagon with an arbitrary triangular hole. He pointed out that a certain ratio corresponding to two such regions has a nice product formula. In this paper, ... More

A Linear Algorithm for Minimum Dominator Colorings of Orientations of PathsJun 11 2019In this paper we present an algorithm for finding a minimum dominator coloring of orientations of paths. To date this is the first algorithm for dominator colorings of digraphs in any capacity. We prove that the algorithm always provides a minimum dominator ... More

Two-dimensional partial cubesJun 11 2019Jun 12 2019We investigate the structure of two-dimensional partial cubes, i.e., of isometric subgraphs of hypercubes whose vertex set defines a set family of VC-dimension at most 2. Equivalently, those are the partial cubes which are not contractible to the 3-cube ... More

Two-dimensional partial cubesJun 11 2019We investigate the structure of two-dimensional partial cubes, i.e., of isometric subgraphs of hypercubes whose vertex set defines a set family of VC-dimension at most 2. Equivalently, those are the partial cubes which are not contractible to the 3-cube ... More

Rearrangement operations on unrooted phylogenetic networksJun 11 2019Rearrangement operations transform a phylogenetic tree into another one and hence induce a metric on the space of phylogenetic trees. Popular operations for unrooted phylogenetic trees are NNI (nearest neighbour interchange), SPR (subtree prune and regraft), ... More

Symmetric multisets of permutationsJun 11 2019The following long-standing problem in combinatorics was first posed in 1993 by Gessel and Reutenauer. For which multisubsets $B$ of the symmetric group $\fS_n$ is the quasisymmetric function $$Q(B) = \sum_{\pi \in B}F_{\Des(\pi), n}$$ a symmetric function? ... More

A Method to construct all the Paving Matroids over a Finite SetJun 11 2019We give a characterization of a matroid to be paving, through its set of hyperplanes and give an algorithm to construct all of them.

Indecomposable $0$-Hecke modules for extended Schur functionsJun 11 2019The extended Schur functions form a basis of quasisymmetric functions that contains the Schur functions. We provide a representation-theoretic interpretation of this basis by constructing $0$-Hecke modules whose quasisymmetric characteristics are the ... More

Resistance distance-based graph invariants and spanning trees of graphs derived from the strong product of $P_2$ and $C_n$Jun 11 2019Let $G_n$ be a graph obtained by the strong product of $P_2$ and $C_n$, where $n\geqslant3$. In this paper, explicit expressions for the Kirchhoff index, multiplicative degree-Kirchhoff index and number of spanning trees of $G_n$ are determined, respectively. ... More

Equitable factorizations of edge-connected graphsJun 10 2019In this paper, we show that a $(3k-3)$-edge-connected graph $G$, under a certain condition on whose degrees, can be edge-decomposed into $k$ factors $G_1,\ldots, G_k$ such that for each vertex $v\in V(G_i)$, $|d_{G_i}(v)-d_G(v)/k|< 1$, where $1\le i\le ... More

Linear recurrences indexed by $\mathbb{Z}$Jun 10 2019This note collects several general results about linear recurrences (also called linear difference equations) in unknowns indexed by the integers. We characterize a unique \emph{reduced} system equivalent to a given linear recurrence, and construct a ... More

Exponential lower bound for Berge-Ramsey problemsJun 10 2019We give an exponential lower bound for Berge-Ramsey problems.

Independence in Arithmetic: The Method of $(\mathcal L, n)$-ModelsJun 10 2019I develop in depth the machinery of $(\mathcal L, n)$-models originally introduced by Shelah \cite{ShelahPA} and, independently in a slightly different form by Kripke (cf \cite{put2000}, \cite{quin80}). This machinery allows fairly routine constructions ... More

Schur ring and Codes for $S$-subgroups over $\Z_{2}^{n}$Jun 10 2019In this paper the relationship between $S$-subgroups in $\Z_{2}^{n}$ and binary codes is shown. If the codes used are both $P(T)$-codes and $G$-codes, then the $S$-subgroup is free. The codes constructed are cyclic, decimated or symmetric and the $S$-subgroups ... More

Abelian tropical coversJun 10 2019The goal of this article is to classify unramified covers of a fixed tropical base curve $\Gamma$ with an action of a finite abelian group G that preserves and acts transitively on the fibers of the cover. We introduce the notion of dilated cohomology ... More

Colored Vertex Models and Iwahori Whittaker FunctionsJun 10 2019We give a recursive method for computing all values of a basis of Iwahori Whittaker functions on a split reductive group $G$ over a nonarchimedean local field $F$ using an action of the Hecke algebra. Then we specialize to $G=GL_r$ where we show that ... More

A note on Lefschetz spheres and their relativesJun 10 2019Inspired by the works of Adiprasito, Babson, Nevo, and Murai on the $g$-conjecture, we consider different classes of PL-spheres and the relations between them. We focus on a certain class of spheres that is in the intersection of vertex-decomposable spheres ... More

Efficient enumeration of non-isomorphic interval graphsJun 10 2019Recently, Yamazaki et al. provided an algorithm that enumerates all non-isomorphic interval graphs on $n$ vertices with an $O(n^6)$ time delay. In this paper, we improve their algorithm and achieve $O(n^3 \log n)$ time delay. We also extend the catalog ... More

More non-bipartite forcing pairsJun 10 2019We study pairs of graphs (H_1,H_2) such that every graph with the densities of H_1 and H_2 close to the densities of H_1 and H_2 in a random graph is quasirandom; such pairs (H_1,H_2) are called forcing. Non-bipartite forcing pairs were first discovered ... More

The extremal number of the subdivisions of the complete bipartite graphJun 10 2019For a graph $F$, the $k$-subdivision of $F$, denoted $F^k$, is the graph obtained by replacing the edges of $F$ with internally vertex-disjoint paths of length $k$. In this paper, we prove that $\mathrm{ex}(n,K_{s,t}^k)=O(n^{1+\frac{s-1}{sk}})$, which ... More

A Dijkstra-Based Efficient Algorithm for Finding a Shortest Non-zero Path in Group-Labeled GraphsJun 10 2019The parity constrained shortest path problem is a well-known variant that is tractable via weighted matching if a graph is undirected and has nonnegative edge length. As a generalization, we focus on the problem of finding a shortest non-zero path between ... More

On the Odd Cycle Game and Connected RulesJun 10 2019We study the positional game where two players, Maker and Breaker, alternately select respectively $1$ and $b$ previously unclaimed edges of $K_n$. Maker wins if she succeeds in claiming all edges of some odd cycle in $K_n$ and Breaker wins otherwise. ... More

Tropical Representations of Plactic MonoidsJun 10 2019We exhibit a faithful representation of the plactic monoid of every finite rank as a monoid of upper triangular matrices over the tropical semiring. This answers a question first posed by Izhakian and subsequently studied by several authors. A consequence ... More

A generalization of Heffter arraysJun 10 2019In this paper we define a new class of partially filled arrays, called relative Heffter arrays, that are a generalization of the Heffter arrays introduced by Archdeacon in 2015. Let $v=2nk+t$ be a positive integer, where $t$ divides $2nk$, and let $J$ ... More

Big Ramsey degrees of 3-uniform hypergraphsJun 10 2019Given a countably infinite hypergraph $\mathcal R$ and a finite hypergraph $\mathcal A$, the big Ramsey degree of $\mathcal A$ in $\mathcal R$ is the least number $L$ such that, for every finite $k$ and every $k$-colouring of the embeddings of $\mathcal ... More

Randomization and reweighted $\ell_1$-minimization for A-optimal design of linear inverse problemsJun 10 2019We consider optimal design of PDE-based Bayesian linear inverse problems with infinite-dimensional parameters. We focus on the A-optimal design criterion, defined as the average posterior variance and quantified by the trace of the posterior covariance ... More

On $2\times 2$ Tropical Commuting MatricesJun 10 2019This paper investigates the geometric properties of the a special case of the two-sided system given by $2 \times 2$ tropical commuting constraints. Given a finite matrix $A \in \mathbb{R}^{2\times 2}$, the paper studies the extreme vertices of the tropical ... More

A Shuffling Theorem for Centrally Symmetric TilingsJun 10 2019In arXiv:1905.08311, Rohatgi and the author proved a shuffling theorem for lozenge tilings of doubly-dented hexagons. The theorem can be considered as a hybrid between two classical theorems in the enumeration of tilings: MacMahon's theorem about centrally ... More

A note on Hedetniemi's conjecture, Stahl's conjecture and the Poljak-Rödl functionJun 10 2019We prove that $\min\{\chi(G), \chi(H)\} - \chi(G\times H)$ can be arbitrarily large, and that if Stahl's conjecture on the multichromatic number of Kneser graphs holds, then $\min\{\chi(G), \chi(H)\}/\chi(G\times H) \leq 1/2 + \epsilon$ for large values ... More

Reconstructing $d$-manifold subcomplexes of cubes from their $(\lfloor d/2 \rfloor + 1)$-skeletonsJun 09 2019In 1984, Dancis proved that any $d$-dimensional simplicial manifold is determined by its $(\lfloor d/2 \rfloor + 1)$-skeleton. This paper adapts his proof to the setting of cubical complexes that can be embedded into a cube of arbitrary dimension. Under ... More

Borders, Palindrome Prefixes, and Square PrefixesJun 09 2019We show that the number of length-$n$ words over a $k$-letter alphabet having no even palindromic prefix is the same as the number of length-$n$ unbordered words, by constructing an explicit bijection between the two sets. A similar result holds for those ... More

The A.B.C.Ds of Schubert calculusJun 09 2019We collect Atiyah-Bott Combinatorial Dreams (A.B.C.Ds) in Schubert calculus. One result relates equivariant structure coefficients for two isotropic flag manifolds, with consequences to the thesis of C. Monical. We contextualize using work of N. Bergeron-F. ... More

Concentration inequalities in spaces of random configurations with positive Ricci curvaturesJun 09 2019In this paper, we prove an Azuma-Hoeffding-type inequality in several classical models of random configurations, including the Erd\H{o}s-R\'enyi random graph models $G(n,p)$ and $G(n,M)$, the random $d$-out(in)-regular directed graphs, and the space of ... More

Positroid varieties and cluster algebrasJun 08 2019We show that the coordinate ring of an open positroid variety coincides with the cluster algebra associated to a Postnikov diagram. This confirms conjectures of Postnikov, Muller--Speyer, and Leclerc, and generalizes results of Scott and Serhiyenko--Sherman-Bennett--Williams. ... More

Extremal problems for convex geometric hypergraphs and ordered hypergraphsJun 08 2019An ordered hypergraph is a hypergraph whose vertex set is linearly ordered, and a convex geometric hypergraph is a hypergraph whose vertex set is cyclically ordered. Extremal problems for ordered and convex geometric graphs have a rich history with applications ... More

A splitting theorem for ordered hypergraphsJun 07 2019An {\em ordered $r$-graph} is an $r$-uniform hypergraph whose vertex set is linearly ordered; it is $r$-{\em interval-partite} if there are $r$ consecutive intervals such that each edge has one point in each interval. The basic observation of Erd\H{o}s ... More

Manifold Matching ComplexesJun 07 2019The matching complex of a graph is the simplicial complex whose vertex set is the set of edges of the graph with a face for each independent set of edges. In this paper we completely characterize the pairs (graph, matching complex) for which the matching ... More

Vandermondes in superspaceJun 07 2019Superspace of rank $n$ is a $\mathbb{Q}$-algebra with $n$ commuting generators $x_1, \dots, x_n$ and $n$ anticommuting generators $\theta_1, \dots, \theta_n$. We present an extension of the Vandermonde determinant to superspace which depends on a sequence ... More

A conjecture of Verstraëte on vertex-disjoint cyclesJun 07 2019Answering a question of H\"aggkvist and Scott, Verstra\"ete proved that every sufficiently large graph with average degree at least $k^2+19k+10$ contains $k$ vertex-disjoint cycles of consecutive even lengths. He further conjectured that the same holds ... More

Singularities and radical initial idealsJun 07 2019What kind of reduced monomial schemes can be obtained as a Gr\"obner degeneration of a smooth projective variety? Our conjectured answer is: only Stanley-Reisner schemes associated to acyclic Cohen-Macaulay simplicial complexes. This would imply, in particular, ... More