arXiv subjects

Alexander Engström

Publications and source records attributed to Alexander Engström.

14 recordsLinked to original sources

arXiv2025

Chromatic numbers from edge ideals: Graph classes with vanishing syzygies are polynomially $\chi$-bounded

The chromatic number $\chi$ of a graph is bounded from below by its clique number $\omega,$ but it can be arbitrary large. Perfect graphs are defined by $\chi=\omega$ for all induced subgraphs. An interesting relaxation are $\chi$-bounded graph classes, where $\chi\leq f(\omega).$ It is not always possible to achieve this with a polynomial $f.$ The edge ideal $I_G$ of a graph $G$ is generated by monomials $x_ux_v$ for each edge $uv$ of $G.$ The bi-graded betti numbers $\beta_{i,j}(I)$ are central algebraic geometric invariants. We study the graph classes where for some fixed $i,j$ that syzygy vanishes, that is, $\beta_{i,j}(I_G)=0.$ We prove that $\chi\leq f(\omega),$ where $f$ is a polynomial of degree $2j-2i-4.$ For the elementary special case $\beta_{i,2i+2}(I_G)=0,$ this amounts to that $(i+1)K_2$-free graphs are ${\omega-1+2i \choose 2i}$-colorable, improving on an old combinatorial result by Wagon. We also show that triangle-free graphs with $\beta_{i,j}(I_G)=0$ are $(j-1)$-colorable. Complexity wise, we show that these colorings can be derived in time $O(n^3)$ for graphs on $n$ vertices. Moreover, we show that for almost all graphs with parabolic $i,j,$ there are better bounds on $\chi.$

math.CO↗

arXiv2023

The regularity of almost all edge ideals

A fruitful contemporary paradigm in graph theory is that almost all graphs that do not contain a certain subgraph have common structural characteristics. The "almost" is crucial, without it there is no structure. In this paper we transfer this paradigm to commutative algebra and make use of deep graph theoretic results. A key tool are the critical graphs introduced by Balogh and Butterfield. We consider edge ideals $I_G$ of graphs and their Betti numbers. The numbers of the form $β_{i,2i+2}$ constitute the "main diagonal" of the Betti table. It is well known that any Betti number $β_{i,j}(I_G)$ below (or equivalently, to the left of) this diagonal is always zero. We identify a certain "parabola" inside the Betti table and call parabolic Betti numbers the entries of the Betti table bounded on the left by the main diagonal and on the right by this parabola. Let $β_{i,j}$ be a parabolic Betti number on the $r$-th row of the Betti table, for $r\ge3$. Our main results state that almost all graphs $G$ with $β_{i,j}(I_G)=0$ can be partitioned into $r-2$ cliques and one independent set, and in particular for almost all graphs $G$ with $β_{i,j}(I_G)=0$ the regularity of $I_G$ is $r-1$.

math.AC↗

arXiv2021

Graded Expectations: Betti numbers and anti-lecture hall compositions of random threshold graphs

This paper examines the one-to-one-to-one correspondence between threshold graphs, Betti numbers of quotients of polynomial rings by $2$-linear ideals, and anti-lecture hall compositions. In particular, we establish new explicit combinatorial mappings between each of these classes of objects and calculate the expected values of the Betti numbers and anti-lecture hall composition corresponding to a random threshold graph.

math.CO↗

arXiv2019

Standard complexes of matroids and lattice paths

Motivated by Gröbner basis theory for finite point configurations, we define and study the class of "standard complexes" associated to a matroid. Standard complexes are certain subcomplexes of the independence complex that are invariant under matroid duality. For the lexicographic term order, the standard complexes satisfy a deletion-contraction-type recurrence. We explicitly determine the lexicographic standard complexes for lattice path matroids using classical bijective combinatorics.

math.CO↗

arXiv2019

Maximum Number of Modes of Gaussian Mixtures

Gaussian mixture models are widely used in Statistics. A fundamental aspect of these distributions is the study of the local maxima of the density, or modes. In particular, it is not known how many modes a mixture of $k$ Gaussians in $d$ dimensions can have. We give a brief account of this problem's history. Then, we give improved lower bounds and the first upper bound on the maximum number of modes, provided it is finite.

math.ST↗

arXiv2018

Transfer-Matrix Methods meet Ehrhart Theory

Transfer-Matrix Methods originated in physics where they were used to count the number of allowed particle states on a structure whose width $n$ is a parameter. Typically, the number of states is exponential in $n.$ One more mathematical instance of this methodology is to enumerate the proper vertex colorings of a graph of growing size by a fixed number of colors. In Ehrhart theory, lattice points in the dilation of a fixed polytope by a factor $k$ are enumerated. By inclusion-exclusion, relevant conditions on how the lattice points interact with hyperplanes are enforced. Typically, the number of points are (quasi-) polynomial in $k.$ The text-book example is that for a fixed graph, the number of proper vertex colorings with $k$ colors is polynomial in $k.$ This paper investigates the joint enumeration problem with both parameters $n$ and $k$ free. We start off with the classical graph colorings and then explore the common situations in combinatorics related to Ehrhart theory. We show how symmetries can be explored to reduce calculations and explain the interactions with Discrete Geometry.

math.CO↗

arXiv2015

Geometric juggling with q-analogues

We derive a combinatorial equilibrium for bounded juggling patterns with a random, $q$-geometric throw distribution. The dynamics are analyzed via rook placements on staircase Ferrers boards, which leads to a steady-state distribution containing $q$-rook polynomial coefficients and $q$-Stirling numbers of the second kind. We show that the equilibrium probabilities of the bounded model can be uniformly approximated with the equilibrium probabilities of a corresponding unbounded model. This observation leads to new limit formulae for $q$-analogues. Keywords: juggling pattern; $q$-Stirling number of the second kind; Ferrers board; Markov process; combinatorial equilibrium

math.CO↗

arXiv2013

Tverberg's theorem and graph coloring

The topological Tverberg theorem has been generalized in several directions by setting extra restrictions on the Tverberg partitions. Restricted Tverberg partitions, defined by the idea that certain points cannot be in the same part, are encoded with graphs. When two points are adjacent in the graph, they are not in the same part. If the restrictions are too harsh, then the topological Tverberg theorem fails. The colored Tverberg theorem corresponds to graphs constructed as disjoint unions of small complete graphs. Hell studied the case of paths and cycles. In graph theory these partitions are usually viewed as graph colorings. As explored by Aharoni, Haxell, Meshulam and others there are fundamental connections between several notions of graph colorings and topological combinatorics. For ordinary graph colorings it is enough to require that the number of colors q satisfy q>Delta, where Delta is the maximal degree of the graph. It was proven by the first author using equivariant topology that if q>Δ^2 then the topological Tverberg theorem still works. It is conjectured that q>KΔis also enough for some constant K, and in this paper we prove a fixed-parameter version of that conjecture. The required topological connectivity results are proven with shellability, which also strengthens some previous partial results where the topological connectivity was proven with the nerve lemma.

math.CO↗

arXiv2013

A nerve lemma for gluing together incoherent discrete Morse functions

Two of the most useful tools in topological combinatorics are the nerve lemma and discrete Morse theory. In this note we introduce a theorem that interpolates between them and allows decompositions of complexes into non-contractible pieces as long as discrete Morse theory ensures that they behave well enough. The proof is based on diagrams of spaces, but that theory is not needed for the formulation or applications of the theorem.

math.CO↗

arXiv2013

Betti diagrams from graphs

The emergence of Boij-Söderberg theory has given rise to new connections between combinatorics and commutative algebra. Herzog, Sharifan, and Varbaro recently showed that every Betti diagram of an ideal with a k-linear minimal resolution arises from that of the Stanley-Reisner ideal of a simplicial complex. In this paper, we extend their result for the special case of 2-linear resolutions using purely combinatorial methods. Specifically, we show bijective correspondences between Betti diagrams of ideals with 2-linear resolutions, threshold graphs, and anti-lecture hall compositions. Moreover, we prove that any Betti diagram of a module with a 2-linear resolution is realized by a direct sum of Stanley-Reisner rings associated to threshold graphs. Our key observation is that these objects are the lattice points in a normal reflexive lattice polytope.

math.CO↗

arXiv2012

Toric Cubes

A toric cube is a subset of the standard cube defined by binomial inequalities. These basic semialgebraic sets are precisely the images of standard cubes under monomial maps. We study toric cubes from the perspective of topological combinatorics. Explicit decompositions as CW-complexes are constructed. Their open cells are interiors of toric cubes and their boundaries are subcomplexes. The motivating example of a toric cube is the edge-product space in phylogenetics, and our work generalizes results known for that space.

math.CO↗

arXiv2011

Polytopes from Subgraph Statistics

Polytopes from subgraph statistics are important in applications and conjectures and theorems in extremal graph theory can be stated as properties of them. We have studied them with a view towards applications by inscribing large explicit polytopes and semi-algebraic sets when the facet descriptions are intractable. The semi-algebraic sets called curvy zonotopes are introduced and studied using graph limits. From both volume calculations and algebraic descriptions we find several interesting conjectures.

math.CO↗

arXiv2005

Independence complexes of claw-free graphs

We study the class of independence complexes of claw-free graphs. The main theorem give good bounds on the connectivity of these complexes, given bounds for a few subcomplexes of the same class. Two applications are presented. Firstly, we show that the independence complex of a claw-free graph with n vertices and maximal degree d is (cn/d+epsilon)-connected, where c=2/3. This can be compared with the result of Szabo and Tardos that c=1/2 is optimal with no restrictions on the graphs. Secondly, we calculate the connectivity of a family of complexes used in Babson and Kozlov's proof of Lovasz conjecture.

math.CO↗

arXiv2005

Transitive graphs in counterexamples to Karp's conjecture

Karp conjectured that all nontrivial monotone graph properties are evasive. This was proved for n a prime power, and n=6, where n is the number of graph vertices, by Kahn, Saks, and Sturtevant. We give a complete description of which transitive graphs are contained in a possible counterexample when n=10.

math.CO↗