arXiv subjects

Raphael Steiner

Publications and source records attributed to Raphael Steiner.

89 records · Page 5Linked to original sources

arXiv2021

Complete minors in digraphs with given dichromatic number

The dichromatic number $\vecχ(D)$ of a digraph $D$ is the smallest $k$ for which it admits a $k$-coloring where every color class induces an acyclic subgraph. Inspired by Hadwiger's conjecture for undirected graphs, several groups of authors have recently studied the containment of directed graph minors in digraphs with given dichromatic number. In this short note we improve several of the existing bounds and prove almost linear bounds by reducing the problem to a recent result of Postle on Hadwiger's conjecture.

math.CO↗

arXiv2020

Disjoint cycles with length constraints in digraphs of large connectivity or minimum degree

A conjecture by Lichiardopol states that for every $k \ge 1$ there exists an integer $g(k)$ such that every digraph of minimum out-degree at least $g(k)$ contains $k$ vertex-disjoint directed cycles of pairwise distinct lengths. Motivated by Lichiardopol's conjecture, we study the existence of vertex-disjoint directed cycles satisfying length constraints in digraphs of large connectivity or large minimum degree. Our main result is that for every $k \in \mathbb{N}$, there exists $s(k) \in \mathbb{N}$ such that every strongly $s(k)$-connected digraph contains $k$ vertex-disjoint directed cycles of pairwise distinct lengths. In contrast, for every $k \in \mathbb{N}$ we construct a strongly $k$-connected digraph containing no two vertex- or arc-disjoint directed cycles of the same length. It is an open problem whether $g(3)$ exists. Here we prove the existence of an integer $K$ such that every digraph of minimum out- and in-degree at least $K$ contains $3$ vertex-disjoint directed cycles of pairwise distinct lengths.

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arXiv2020

Even Circuits in Oriented Matroids

In this paper we generalise the even directed cycle problem, which asks whether a given digraph contains a directed cycle of even length, to orientations of regular matroids. We define non-even oriented matroids generalising non-even digraphs, which played a central role in resolving the computational complexity of the even dicycle problem. Then we show that the problem of detecting an even directed circuit in a regular matroid is polynomially equivalent to the recognition of non-even oriented matroids. Our main result is a precise characterisation of the class of non-even oriented bond matroids in terms of forbidden minors, which complements an existing characterisation of non-even oriented graphic matroids by Seymour and Thomassen.

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arXiv2020

Oriented cycles in digraphs of large outdegree

In 1985, Mader conjectured that for every acyclic digraph $F$ there exists $K=K(F)$ such that every digraph $D$ with minimum out-degree at least $K$ contains a subdivision of $F$. This conjecture remains widely open, even for digraphs $F$ on five vertices. Recently, Aboulker, Cohen, Havet, Lochet, Moura and Thomassé studied special cases of Mader's problem and made the following conjecture: for every $\ell \geq 2$ there exists $K = K(\ell)$ such that every digraph $D$ with minimum out-degree at least $K$ contains a subdivision of every orientation of a cycle of length $\ell$. We prove this conjecture and answer further open questions raised by Aboulker et al.

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arXiv2020

Dichromatic number and forced subdivisions

We investigate bounds on the dichromatic number of digraphs which avoid a fixed digraph as a topological minor. For a digraph $F$, denote by $\text{mader}_{\vecχ}(F)$ the smallest integer $k$ such that every $k$-dichromatic digraph contains a subdivision of $F$. As our first main result, we prove that if $F$ is an orientation of a cycle then $\text{mader}_{\vecχ}(F)=v(F)$. This settles a conjecture of Aboulker, Cohen, Havet, Lochet, Moura and Thomassé. We also extend this result to the more general class of orientations of cactus graphs, and to bioriented forests. Our second main result is that $\text{mader}_{\vecχ}(F)=4$ for every tournament $F$ of order $4$. This is an extension of the classical result by Dirac that $4$-chromatic graphs contain a $K_4$-subdivision to directed graphs.

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arXiv2020

Pentagon contact representations

Representations of planar triangulations as contact graphs of a set of internally disjoint homothetic triangles or of a set of internally disjoint homothetic squares have received quite some attention in recent years. In this paper we investigate representations of planar triangulations as contact graphs of a set of internally disjoint homothetic pentagons. Surprisingly such a representation exists for every triangulation whose outer face is a 5-gon. We relate these representations to five color forests. These combinatorial structures resemble Schnyder woods and transversal structures, respectively. In particular there is a bijection to certain alpha-orientations and consequently a lattice structure on the set of five color forests of a given graph. This lattice structure plays a role in an algorithm that is supposed to compute a contact representation with pentagons for a given graph. Based on a five color forest the algorithm builds a system of linear equations and solves it, if the solution is non-negative, it encodes distances between corners of a pentagon representation. In this case the representation is constructed and the algorithm terminates. Otherwise negative variables guide a change of the five color forest and the procedure is restarted with the new five color forest. Similar algorithms have been proposed for contact representations with homothetic triangles and with squares.

cs.CG↗

arXiv2020

A Note on Coloring Digraphs of Large Girth

The digirth of a digraph is the length of a shortest directed cycle. The dichromatic number $\vecχ(D)$ of a digraph $D$ is the smallest size of a partition of the vertex-set into subsets inducing acyclic subgraphs. A conjecture by Harutyunyan and Mohar states that $\vecχ(D) \le \left\lceil\fracΔ{4}\right\rceil+1$ for every digraph $D$ of digirth at least $3$ and maximum degree $Δ$. The best known partial result by Golowich shows that $\vecχ(D) \le \frac{2}{5}Δ+O(1)$. In this short note we prove for every $g \ge 2$ that if $D$ is a digraph of digirth at least $2g-1$ and maximum degree $Δ$, then $\vecχ(D) \le (\frac{1}{3}+\frac{1}{3g}) Δ+ O_g(1)$. This improves the bound of Golowich for digraphs without directed cycles of length at most $10$.

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arXiv2020

On the Average Complexity of the $k$-Level

Let ${\cal L}$ be an arrangement of $n$ lines in the Euclidean plane. The \emph{$k$-level} of ${\cal L}$ consists of all vertices $v$ of the arrangement which have exactly $k$ lines of ${\cal L}$ passing below $v$. The complexity (the maximum size) of the $k$-level in a line arrangement has been widely studied. In 1998 Dey proved an upper bound of $O(n\cdot (k+1)^{1/3})$. Due to the correspondence between lines in the plane and great-circles on the sphere, the asymptotic bounds carry over to arrangements of great-circles on the sphere, where the $k$-level denotes the vertices at distance at most $k$ to a marked cell, the \emph{south pole}. We prove an upper bound of $O((k+1)^2)$ on the expected complexity of the $k$-level in great-circle arrangements if the south pole is chosen uniformly at random among all cells. We also consider arrangements of great $(d-1)$-spheres on the sphere $\mathbb{S}^d$ which are orthogonal to a set of random points on $\mathbb{S}^d$. In this model, we prove that the expected complexity of the $k$-level is of order $Θ((k+1)^{d-1})$.

cs.CG↗

arXiv2020

On the Complexity of Digraph Colourings and Vertex Arboricity

It has been shown by Bokal et al. that deciding 2-colourability of digraphs is an NP-complete problem. This result was later on extended by Feder et al. to prove that deciding whether a digraph has a circular $p$-colouring is NP-complete for all rational $p>1$. In this paper, we consider the complexity of corresponding decision problems for related notions of fractional colourings for digraphs and graphs, including the star dichromatic number, the fractional dichromatic number and the circular vertex arboricity. We prove the following results: Deciding if the star dichromatic number of a digraph is at most $p$ is NP-complete for every rational $p>1$. Deciding if the fractional dichromatic number of a digraph is at most $p$ is NP-complete for every $p>1, p \neq 2$. Deciding if the circular vertex arboricity of a graph is at most $p$ is NP-complete for every rational $p>1$. To show these results, different techniques are required in each case. In order to prove the first result, we relate the star dichromatic number to a new notion of homomorphisms between digraphs, called circular homomorphisms, which might be of independent interest. We provide a classification of the computational complexities of the corresponding homomorphism colouring problems similar to the one derived by Feder et al. for acyclic homomorphisms.

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arXiv2019

Majority Colorings of Sparse Digraphs

A majority coloring of a directed graph is a vertex-coloring in which every vertex has the same color as at most half of its out-neighbors. Kreutzer, Oum, Seymour, van der Zypen and Wood proved that every digraph has a majority 4-coloring and conjectured that every digraph admits a majority 3-coloring. We verify this conjecture for digraphs with chromatic number at most 6 or dichromatic number at most 3. We obtain analogous results for list coloring: We show that every digraph with list chromatic number at most 6 or list dichromatic number at most 3 is majority 3-choosable. We deduce that digraphs with maximum out-degree at most 4 or maximum degree at most 7 are majority 3-choosable. On the way to these results we investigate digraphs admitting a majority 2-coloring. We show that every digraph without odd directed cycles is majority 2-choosable. We answer an open question posed by Kreutzer et al. negatively, by showing that deciding whether a given digraph is majority 2-colorable is NP-complete. Finally we deal with a fractional relaxation of majority coloring proposed by Kreutzer et al. and show that every digraph has a fractional majority 3.9602-coloring. We show that every digraph with minimum out-degree $Ω\left((1/\varepsilon)^2\ln(1/\varepsilon)\right)$ has a fractional majority $(2+\varepsilon)$-coloring.

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arXiv2019

A Note On Universal Point Sets for Planar Graphs

We investigate which planar point sets allow simultaneous straight-line embeddings of all planar graphs on a fixed number of vertices. We first show that $(1.293-o(1))n$ points are required to find a straight-line drawing of each $n$-vertex planar graph (vertices are drawn as the given points); this improves the previous best constant $1.235$ by Kurowski (2004). Our second main result is based on exhaustive computer search: We show that no set of 11 points exists, on which all planar 11-vertex graphs can be simultaneously drawn plane straight-line. This strengthens the result by Cardinal, Hoffmann, and Kusters (2015), that all planar graphs on $n \le 10$ vertices can be simultaneously drawn on particular `universal' sets of $n$ points while there are no universal sets for $n \ge 15$. Moreover, we provide a set of 49 planar 11-vertex graphs which cannot be simultaneously drawn on any set of 11 points. This, in fact, is another step towards a (negative) answer of the question, whether every two planar graphs can be drawn simultaneously -- a question raised by Brass, Cenek, Duncan, Efrat, Erten, Ismailescu, Kobourov, Lubiw, and Mitchell (2007).

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arXiv2019

A Note on Graphs of Dichromatic Number 2

Neumann-Lara and Škrekovski conjectured that every planar digraph is $2$-colourable. We show that this conjecture is equivalent to the more general statement that all oriented $K_5$-minor-free graphs are $2$-colourable.

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arXiv2019

Parametrised Algorithms for Directed Modular Width

Many well-known NP-hard algorithmic problems on directed graphs resist efficient parametrisations with most known width measures for directed graphs, such as directed treewidth, DAG-width, Kelly-width and many others. While these focus on measuring how close a digraph is to an oriented tree resp. a directed acyclic graph, in this paper, we investigate directed modular width as a parameter, which is closer to the concept of clique-width. We investigate applications of modular decompositions of directed graphs to a wide range of algorithmic problems and derive FPT-algorithms for several well-known digraph-specific NP-hard problems, namely minimum (weight) directed feedback vertex set, minimum (weight) directed dominating set, digraph colouring, directed Hamiltonian path/cycle, partitioning into paths, (capacitated) vertex-disjoint directed paths, and the directed subgraph homeomorphism problem. The latter yields a polynomial-time algorithm for detecting topological minors in digraphs of bounded directed modular width. Finally we illustrate that also other structural digraph parameters, such as the directed pathwidth and the cycle-rank can be computed efficiently using directed modular width as a parameter.

cs.DM↗

arXiv2019

Complete Acyclic Colorings

We study two parameters that arise from the dichromatic number and the vertex-arboricity in the same way that the achromatic number comes from the chromatic number. The adichromatic number of a digraph is the largest number of colors its vertices can be colored with such that every color induces an acyclic subdigraph but merging any two colors yields a monochromatic directed cycle. Similarly, the a-vertex arboricity of an undirected graph is the largest number of colors that can be used such that every color induces a forest but merging any two yields a monochromatic cycle. We study the relation between these parameters and their behavior with respect to other classical parameters such as degeneracy and most importantly feedback vertex sets.

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arXiv2019

Colouring Non-Even Digraphs

A colouring of a digraph as defined by Erdos and Neumann-Lara in 1980 is a vertex-colouring such that no monochromatic directed cycles exist. The minimal number of colours required for such a colouring of a loopless digraph is defined to be its dichromatic number. This quantity has been widely studied in the last decades and can be considered as a natural directed analogue of the chromatic number of a graph. A digraph D is called even if for every 0-1-weighting of the edges it contains a directed cycle of even total weight. We show that every non-even digraph has dichromatic number at most 2 and an optimal colouring can be found in polynomial time. We strengthen a previously known NP-hardness result by showing that deciding whether a directed graph is 2-colourable remains NP-hard even if it contains a feedback vertex set of bounded size.

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arXiv2019

The Star Dichromatic Number

We introduce a new notion of circular colourings for digraphs. The idea of this quantity, called star dichromatic number $\vecχ^\ast(D)$ of a digraph $D$, is to allow a finer subdivision of digraphs with the same dichromatic number into such which are "easier" or "harder" to colour by allowing fractional values. This is related to a coherent notion for the vertex arboricity of graphs introduced by Wang et al. and resembles the concept of the star chromatic number of graphs introduced by Vince in the framework of digraph colouring. After presenting basic properties of the new quantity, including range, simple classes of digraphs, general inequalities and its relation to integer counterparts as well as other concepts of fractional colouring, we compare our notion with the notion of circular colourings for digraphs introduced by Bokal et al. and point out similarities as well as differences in certain situations. As it turns out, the star dichromatic number is a lower bound for the circular dichromatic number of Bokal et al., but the gap between the numbers may be arbitrarily close to $1$. We conclude with a discussion of the case of planar digraphs and point out some open problems.

math.CO↗

arXiv2019

Flip distances between graph orientations

Flip graphs are a ubiquitous class of graphs, which encode relations induced on a set of combinatorial objects by elementary, local changes. Skeletons of associahedra, for instance, are the graphs induced by quadrilateral flips in triangulations of a convex polygon. For some definition of a flip graph, a natural computational problem to consider is the flip distance: Given two objects, what is the minimum number of flips needed to transform one into the other? We consider flip graphs on orientations of simple graphs, where flips consist of reversing the direction of some edges. More precisely, we consider so-called $α$-orientations of a graph $G$, in which every vertex $v$ has a specified outdegree $α(v)$, and a flip consists of reversing all edges of a directed cycle. We prove that deciding whether the flip distance between two $α$-orientations of a planar graph $G$ is at most two is \NP-complete. This also holds in the special case of perfect matchings, where flips involve alternating cycles. This problem amounts to finding geodesics on the common base polytope of two partition matroids, or, alternatively, on an alcoved polytope. It therefore provides an interesting example of a flip distance question that is computationally intractable despite having a natural interpretation as a geodesic on a nicely structured combinatorial polytope. We also consider the dual question of the flip distance betwe en graph orientations in which every cycle has a specified number of forward edges, and a flip is the reversal of all edges in a minimal directed cut. In general, the problem remains hard. However, if we restrict to flips that only change sinks into sources, or vice-versa, then the problem can be solved in polynomial time. Here we exploit the fact that the flip graph is the cover graph of a distributive lattice. This generalizes a recent result from Zhang, Qian, and Zhang.

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