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Kolja Knauer

Publications and source records attributed to Kolja Knauer.

At least 37 records · Page 2Linked to original sources
arXiv2025

On endomorphism universality of sparse graph classes

We show that every commutative idempotent monoid (a.k.a lattice) is the endomorphism monoid of a subcubic graph. This solves a problem of Babai and Pultr [J. Comb.~Theory, Ser.~B, 1980] and the degree bound is best-possible. On the other hand, we show that no class excluding a minor can have all commutative idempotent monoids among its endomorphism monoids. As a by-product we prove that monoids can be represented by graphs of bounded expansion (reproving a result of Nešetřil and Ossona de Mendez) and $k$-cancellative monoids can be represented by graphs of bounded degree. Finally, we show that not all completely regular monoids can be represented by graphs excluding topological minor (strengthening a result of Babai and Pultr).

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arXiv2025

On rigid regular graphs and a problem of Babai and Pultr

A graph is \textit{rigid} if it only admits the identity endomorphism. We show that for every $d\ge 3$ there exist infinitely many mutually rigid $d$-regular graphs of arbitrary odd girth $g\geq 7$. Moreover, we determine the minimum order of a rigid $d$-regular graph for every $d\ge 3$. This provides strong positive answers to a question of van der Zypen [https://mathoverflow.net/q/296483, https://mathoverflow.net/q/321108]. Further, we use our construction to show that every finite monoid is isomorphic to the endomorphism monoid of a regular graph. This solves a problem of Babai and Pultr [J. Comb.~Theory, Ser.~B, 1980].

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arXiv2024

Coloring minimal Cayley graphs

In 1978 Babai raised the question whether all minimal Cayley graphs have bounded chromatic number; in 1994 he conjectured a negative answer. In this paper we show that any minimal Cayley graph of a (finitely generated) generalized dihedral or nilpotent group has chromatic number at most 3, while 4 colors are sometimes necessary for soluble groups. On the other hand we address a related question proposed by Babai in 1978 by constructing graphs of unbounded chromatic number that admit a proper edge coloring such that each cycle has some color at least twice. The latter can be viewed as a step towards confirming Babai's 1994 conjecture -- a problem that remains open.

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arXiv2024

Geometry of convex geometries

We prove that any convex geometry $\mathcal{A}=(U,\mathcal{C})$ on $n$ points and any ideal $\mathcal{I}=(U',\mathcal{C}')$ of $\mathcal{A}$ can be realized as the intersection pattern of an open convex polyhedral cone $K\subseteq {\mathbb R}^n$ with the orthants of ${\mathbb R}^n$. Furthermore, we show that $K$ can be chosen to have at most $m$ facets, where $m$ is the number of critical rooted circuits of $\mathcal{A}$. We also show that any convex geometry of convex dimension $d$ is realizable in ${\mathbb R}^d$ and that any multisimplicial complex (a basic example of an ideal of a convex geometry) of dimension $d$ is realizable in ${\mathbb R}^{2d}$ and that this is best possible. From our results it also follows that distributive lattices of dimension $d$ are realizable in ${\mathbb R}^{d}$ and that median systems are realizable. We leave open %the question whether each median system of dimension $d$ is realizable in ${\mathbb R}^{O(d)}$.

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arXiv2024

Partitioning a Planar Graph into two Triangle-Forests

We show that the vertices of every planar graph can be partitioned into two sets, each inducing a so-called triangle-forest, i.e., a graph with no cycles of length more than three. We further discuss extensions to locally planar graphs. After finishing the paper we noticed that our main result was already proved much earlier by Carsten Thomassen [Decomposing a Planar Graph into Degenerate Graphs, JCTB 1995].

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arXiv2024

Sensitivity of $m$-ary functions and low degree partitions of Hamming graphs

The study of complexity measures of Boolean functions led Nisan and Szegedy to state the sensitivity conjecture in 1994, claiming a polynomial relation between degree and sensitivity. This problem remained unsolved until 2019, when Huang proved the conjecture via an equivalent graph theoretical reformulation due to Gotsman and Linial. We study $m$-ary functions, i.e., functions $f: T^n \rightarrow T$ where $T\subseteq \mathbb{C}$ is a finite alphabet of cardinality $|T| = m $ and extend the notions of degree $\mathrm{deg}(f)$ and sensitivity $s(f)$ to $m$-ary functions and show $s(f)\in O(\mathrm{deg}(f)^2)$. This generalizes results of Nisan and Szegedy. Conversely, we introduce the $m$-ary sensitivity conjecture, claiming a polynomial upper bound for $\mathrm{deg}(f)$ in terms of $s(f)$. Analogously to results of Gotsman and Linial, we provide a formulation of the conjecture in terms of imbalanced partitions of Hamming graphs into low degree subgraphs. Combining this with ideas of Chung, Füredi, Graham and Seymour, we show that for any prime $p$ the bound in the $p$-ary sensitivity conjecture has to be at least quadratic: there exist $p$-ary functions $f$ of arbitrarily large degree and $\mathrm{deg}(f)\in Ω(s(f)^2)$.

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arXiv2024

Finitary affine oriented matroids

We initiate the axiomatic study of affine oriented matroids (AOMs) on arbitrary ground sets, obtaining fundamental notions such as minors, reorientations and a natural embedding into the frame work of Complexes of Oriented Matroids. The restriction to the finitary case (FAOMs) allows us to study tope graphs and covector posets, as well as to view FAOMs as oriented finitary semimatroids. We show shellability of FAOMs and single out the FAOMs that are affinely homeomorphic to $\mathbb{R}^n$. Finally, we study group actions on AOMs, whose quotients in the case of FAOMs are a stepping stone towards a general theory of affine and toric pseudoarrangements. Our results include applications of the multiplicity Tutte polynomial of group actions of semimatroids, generalizing enumerative properties of toric arrangements to a combinatorially defined class of arrangements of submanifolds. This answers partially a question by Ehrenborg and Readdy.

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arXiv2024

On $k$-neighborly reorientations of oriented matroids

We study the existence and the number of $k$-neighborly reorientations of an oriented matroid. This leads to $k$-variants of McMullen's problem and Roudneff's conjecture, the case $k=1$ being the original statements on complete cells in arrangements. Adding to results of Larman and García-Colín, we provide new bounds on the $k$-McMullen's problem and prove the conjecture for several ranks and $k$ by computer. Further, we show that $k$-Roudneff's conjecture for fixed rank and $k$ reduces to a finite case analyse. As a consequence we prove the conjecture for odd rank $r$ and $k=\frac{r-1}{2}$ as well as for rank $6$ and $k=2$ with the aid of the computer.

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arXiv2024

Lattice path matroids and quotients

We characterize the quotients among lattice path matroids (LPMs) in terms of their diagrams. This characterization allows us to show that ordering LPMs by quotients yields a graded poset, whose rank polynomial has the Narayana numbers as coefficients. Furthermore, we study full lattice path flag matroids and show that -- contrary to arbitrary positroid flag matroids -- they correspond to points in the nonnegative flag variety. At the basis of this result lies an identification of certain intervals of the strong Bruhat order with lattice path flag matroids. A recent conjecture of Mcalmon, Oh, and Xiang states a characterization of quotients of positroids. We use our results to prove this conjecture in the case of LPMs.

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arXiv2024

Concepts of Dimension for Convex Geometries

Let $X$ be a finite set. A family $P$ of subsets of $X$ is called a convex geometry with ground set $X$ if (1) $\emptyset, X\in P$; (2) $A\cap B\in P$ whenever $A,B\in P$; and (3) if $A\in P$ and $A\neq X$, there is an element $α\in X-A$ such that $A\cup\{α\}\in P$. As a non-empty family of sets, a convex geometry has a well defined VC-dimension. In the literature, a second parameter, called convex dimension, has been defined expressly for these structures. Partially ordered by inclusion, a convex geometry is also a poset, and four additional dimension parameters have been defined for this larger class, called Dushnik-Miller dimension, Boolean dimension, local dimension, and fractional dimension, espectively. For each pair of these six dimension parameters, we investigate whether there is an infinite class of convex geometries on which one parameter is bounded and the other is not.

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arXiv2023

Plattenbauten: Touching Rectangles in Space

Planar bipartite graphs can be represented as touching graphs of horizontal and vertical segments in $\mathbb{R}^2$. We study a generalization in space: touching graphs of axis-aligned rectangles in $\mathbb{R}^3$, and prove that planar 3-colorable graphs can be represented this way. The result implies a characterization of corner polytopes previously obtained by Eppstein and Mumford. A by-product of our proof is a distributive lattice structure on the set of orthogonal surfaces with given skeleton. Further, we study representations by axis-aligned non-coplanar rectangles in $\mathbb{R}^3$ such that all regions are boxes. We show that the resulting graphs correspond to octahedrations of an octahedron. This generalizes the correspondence between planar quadrangulations and families of horizontal and vertical segments in $\mathbb{R}^2$ with the property that all regions are rectangles.

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arXiv2023

Labeled sample compression schemes for complexes of oriented matroids

We show that the topes of a complex of oriented matroids (abbreviated COM) of VC-dimension $d$ admit a proper labeled sample compression scheme of size $d$. This considerably extends results of Moran and Warmuth on ample classes, of Ben-David and Litman on affine arrangements of hyperplanes, and of the authors on complexes of uniform oriented matroids, and is a step towards the sample compression conjecture -- one of the oldest open problems in computational learning theory. On the one hand, our approach exploits the rich combinatorial cell structure of COMs via oriented matroid theory. On the other hand, viewing tope graphs of COMs as partial cubes creates a fruitful link to metric graph theory.

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arXiv2023

Roudneff's Conjecture in Dimension $4$

J.-P. Roudneff conjectured in 1991 that every arrangement of $n \ge 2d+1\ge 5$ pseudohyperplanes in the real projective space $\mathbb{P}^d$ has at most $\sum_{i=0}^{d-2} \binom{n-1}{i}$ complete cells (i.e., cells bounded by each hyperplane). The conjecture is true for $d=2,3$ and for arrangements arising from Lawrence oriented matroids. The main result of this manuscript is to show the validity of Roudneff's conjecture for $d=4$. Moreover, based on computational data we conjecture that the maximum number of complete cells is only obtained by cyclic arrangements.

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arXiv2023

Corners and simpliciality in oriented matroids and partial cubes

Building on a recent characterization of tope graphs of Complexes of Oriented Matroids (COMs), we tackle and generalize several classical problems in Oriented Matroids (OMs), Lopsided Sets (aka ample set systems), and partial cubes via Metric Graph Theory. Our first main result is that every element of an OM from a class introduced by Mandel is incident to a simplicial tope, i.e, such OMs contain no mutation-free elements. This allows us to refute a conjecture of Mandel from 1983, that would have implied the famous Las Vergnas' simplex conjecture. Further, we show that the mutation graph of uniform OMs of order at most 9 are connected, thus confirming a stronger conjecture of Cordovil-Las Vergnas in this setting. The second main contribution is the introduction of corners of COMs as a natural generalization of corners in Lopsided Sets. Generalizing results of Bandelt and Chepoi, Tracy Hall, and Chalopin et al. we prove that realizable COMs, rank 2 COMs, as well as hypercellular graphs admit corner peelings. Using this, we confirm Las Vergnas' conjecture for antipodal partial cubes (a class much lager than OMs) of small rank or isometric dimension.

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arXiv2022

Kirchberger's Theorem for Complexes of Oriented Matroids

The separation theorem of Kirchberger can be proven using a combination of Farkas' Lemma and Caratheodory's Theorem. Since those theorems are at the heart of oriented matroids, we are interested in a generalization of Kirchberger's Theorem to them. This has already been done for rank 3 oriented matroids. Here we prove it for complexes of oriented matroids, which are a generalization of oriented matroids.

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arXiv2022

Feedback vertex sets in (directed) graphs of bounded degeneracy or treewidth

We study the minimum size $f$ of a feedback vertex set in directed and undirected $n$-vertex graphs of given degeneracy or treewidth. In the undirected setting the bound $\frac{k-1}{k+1}n$ is known to be tight for graphs with bounded treewidth $k$ or bounded odd degeneracy $k$. We show that neither of the easy upper and lower bounds $\frac{k-1}{k+1}n$ and $\frac{k}{k+2}n$ can be exact for the case of even degeneracy. More precisely, for even degeneracy $k$ we prove that $f < \frac{k}{k+2}n$ and for every $ε>0$, there exists a $k$-degenerate graph for which $f\geq \frac{3k-2}{3k+4}n -ε$. For directed graphs of bounded degeneracy $k$, we prove that $f\leq\frac{k-1}{k+1}n$ and that this inequality is strict when $k$ is odd. For directed graphs of bounded treewidth $k\geq 2$, we show that $f \leq \frac{k}{k+3}n$ and for every $ε>0$, there exists a $k$-degenerate graph for which $f\geq \frac{k-2\lfloor\log_2(k)\rfloor}{k+1}n -ε$. Further, we provide several constructions of low degeneracy or treewidth and large $f$.

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arXiv2022

Beyond symmetry in generalized Petersen graphs

A graph is a core or unretractive if all its endomorphisms are automorphisms. Well-known examples of cores include the Petersen graph and the graph of the dodecahedron -- both generalized Petersen graphs. We characterize the generalized Petersen graphs that are cores. A simple characterization of endomorphism-transitive generalized Petersen graphs follows. This extends the characterization of vertex-transitive generalized Petersen graphs due to Frucht, Graver, and Watkins and solves a problem of Fan and Xie. Moreover, we study generalized Petersen graphs that are (underlying graphs of) Cayley graphs of monoids. We show that this is the case for the Petersen graph, answering a recent mathoverflow question, for the Desargues graphs, and for the dodecahedron -- answering a question of Knauer and Knauer. Moreover, we characterize the infinite family of generalized Petersen graphs that are Cayley graph of a monoid with generating connection set of size two. This extends Nedela and Škoviera's characterization of generalized Petersen graphs that are group Cayley graphs and complements results of Hao, Gao, and Luo.

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