arXiv subjects

Kolja Knauer

Publications and source records attributed to Kolja Knauer.

At least 19 recordsLinked to original sources

arXiv2026

Logarithmic basis number of graphs

The basis number $\mathrm{bn}(G)$ of a graph $G$ is the minimum edge-congestion of a basis of its cycle space. We prove that every finite $n$-vertex multigraph satisfies \[ \mathrm{bn}(G)=O(\log n), \] resolving, for simple graphs, a question of Bazargani, Biedl, Bose, Maheshwari and Miraftab, subsequently stated as a conjecture by Miraftab, Morin and Yuditsky. The argument also yields the cycle-rank refinement \[ \mathrm{bn}(G)=O(\log \beta(G)), \] where $\beta(G)$ is the dimension of the cycle space, and a reduction of Lehner and Miraftab, based on a theorem of Richter and Shank, then gives \[ \mathrm{bn}(G)=O(\log g) \] for graphs of Euler genus $g$. These orders are best possible.

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arXiv2026

Coloring semiminimal Cayley Graphs

In 1978 Babai raised the question whether all semiminimal Cayley graphs have bounded chromatic number. In this paper we show that semiminimal Cayley graphs of abelian and generalized dihedral groups have circular chromatic number at most 4, thus extending a result of Barajas and Serra.

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arXiv2026

Embracing exchange sequences and oriented matroid polyhedron diameter

We reduce the embracing exchange distance of bases of oriented matroids to the metric of oriented matroid polyhedra. This allows us to disprove recent conjectures of Caoduro, Khodamoradi, Paat, and Shepherd and of B\'erczi and N\'ador. On the other hand, we show that any two embracing bases of an oriented matroid of rank $r$ can be transformed into each other in at most $2r^{\log_2(r)+3}$ steps and in at most $r$ steps in a graphic oriented matroid or a Lawrence oriented matroid, thus confirming the conjecture in these cases.

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arXiv2025

Hamiltonian connectivity of some base-cobase graphs

There has been wide interest in understanding which properties of base graphs of matroids extend to base-cobase graphs of matroids. A significant result of Naddef and Pulleyblank (1984) shows that the $1$-skeleton of any $(0,1)$-polytope is either a hypercube, or Hamiltonian-connected, i.e. there is a Hamiltonian path connecting any two vertices. In particular, this is true for base graphs of matroids. A natural question raised by Farber, Richter, and Shank (1985) is whether this extends to base-cobase graphs. First, we use the polytopal approach to show Hamiltonian connectivity of base-cobase graphs of series-parallel extensions of lattice path matroids. On the other hand, we show that this method extends to only very special classes related to identically self-dual matroids. Second, we show that base-cobase graphs of wheels and whirls are Hamiltonian connected. Last, we show that the regular matroid $R_{10}$ yields a negative answer to the question of Farber, Richter, and Shank.

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arXiv2025

What is and is not inside a Cayley graph?

In this note we show that there is a cubic graph of girth $5$ that is not a subgraph of any minimal Cayley graph. On the other hand, we show that any Generalized Petersen Graph $G(n,k)$ with $\gcd(n,k)=1$ is an induced subgraph of a minimal Cayley graph. These results give insights into two comments of L\'aszl\'o Babai in [L. Babai, \emph{Automorphism groups, isomorphism, reconstruction}. Graham, R. L. (ed.) et al., Handbook of combinatorics. Vol. 1-2, 1994].

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arXiv2025

Odd coloring graphs with linear neighborhood complexity

We prove that any class of graphs with linear neighborhood complexity has bounded improper odd chromatic number. As a result, if $\mathcal{G}$ is the class of all circle graphs, or if $\mathcal{G}$ is any class with bounded twin-width, bounded merge-width, or a forbidden vertex-minor, then $\mathcal{G}$ is $\chi_{\mathrm{o}}$-bounded.

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arXiv2025

Cell structure of mediangle graphs

Mediangle graphs are a common generalization of median graphs (1-sekeleta of CAT(0) cube complexes) and Coxeter graphs (Cayley graphs of Coxeter systems). Answering a question motivated from geometric group theory, we show that these graphs can be endowed with the structure of a contractible cell complex. We further show that the cells of this complex are products of simplices and simplicial oriented matroids. A crucial part of the proof identifies bipartite mediangle graphs as tope graphs of finitary Complexes of Oriented Matroids.

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arXiv2025

Sensitivity and Hamming graphs

For any $m\geq 3$ we show that the Hamming graph $H(n,m)$ admits an imbalanced partition into $m$ sets, each inducing a subgraph of low maximum degree. This improves previous results by Tandya and by Potechin and Tsang, and disproves the Strong $m$-ary Sensitivity Conjecture of Asensio, Garc\'ia-Marco, and Knauer. On the other hand, we prove their weaker $m$-ary Sensitivity Conjecture by showing that the sensitivity of any $m$-ary function is bounded from below by a polynomial expression in its degree.

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arXiv2025

Strong odd coloring in minor-closed classes

We show that the strong odd chromatic number on any proper minor-closed graph class is bounded by a constant. We almost determine the smallest such constant for outerplanar graphs.

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arXiv2025

Girth in $GF(q)$-representable matroids

We prove a conjecture of Geelen, Gerards, and Whittle that for any finite field $GF(q)$ and any integer $t$, every cosimple $GF(q)$-representable matroid with sufficiently large girth contains either $M(K_t)$ or $M(K_t)^*$ as a minor.

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arXiv2025

Shellability of the quotient order on lattice path matroids

The concept of a matroid quotient has connections to fundamental questions in the geometry of flag varieties. In previous work, Benedetti and Knauer characterized quotients in the class of lattice path matroids (LPMs) in terms of a simple combinatorial condition. As a consequence, they showed that the quotient order on LPMs yields a graded poset whose rank polynomial relates to a refinement of the Catalan numbers. In this work we show that this poset admits an EL-labeling, implying that the order complex is shellable and hence enjoys several combinatorial and topological properties. We use this to establish bounds on the M\"obius function of the poset, interpreting falling chains in the EL-labeling in terms of properties of underlying permutations. Furthermore, we show that this EL-labeling is in fact a Whitney labeling, in the sense of the recent notion introduced by Gonz\'alez D'Le\'on and Hallam.

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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

Partitions of planar (oriented) graphs into a connected acyclic and an independent set

A question at the intersection of Barnette's Hamiltonicity and Neumann-Lara's dicoloring conjecture is: Can every Eulerian oriented planar graph be vertex-partitioned into two acyclic sets? A CAI-partition of an undirected/oriented graph is a partition into a tree/connected acyclic subgraph and an independent set. Consider any plane Eulerian oriented triangulation together with its unique tripartition, i.e. partition into three independent sets. If two of these three sets induce a subgraph G that has a CAI-partition, then the above question has a positive answer. We show that if G is subcubic, then it has a CAI-partition, i.e. oriented planar bipartite subcubic 2-vertex-connected graphs admit CAI-partitions. We also show that series-parallel 2-vertex-connected graphs admit CAI-partitions. Finally, we present a Eulerian oriented triangulation such that no two sets of its tripartition induce a graph with a CAI-partition. This generalizes a result of Alt, Payne, Schmidt, and Wood to the oriented setting.

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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, F\"uredi, 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 $\mathrm{deg}(f)\in \Omega(s(f)^2)$.

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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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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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