arXiv subjects

Raphael Steiner

Publications and source records attributed to Raphael Steiner.

At least 37 records · Page 2Linked to original sources

arXiv2025

Longest cycles in vertex-transitive and highly connected graphs

We present progress on three old conjectures about longest paths and cycles in graphs. The first pair of conjectures, due to Lovász from 1969 and Thomassen from 1978, respectively, states that all connected vertex-transitive graphs contain a Hamiltonian path, and that all sufficiently large such graphs even contain a Hamiltonian cycle. The third conjecture, due to Smith from 1984, states that for $r\ge 2$ in every $r$-connected graph any two longest cycles intersect in at least $r$ vertices. In this paper, we prove a new lemma about the intersection of longest cycles in a graph which can be used to improve the best known bounds towards all the aforementioned conjectures: First, we show that every connected vertex-transitive graph on $n\geq 3$ vertices contains a cycle (and hence path) of length at least $Ω(n^{13/21})$, improving on $Ω(n^{3/5})$ from [DeVos, \emph{arXiv:2302:04255}, 2023]. Second, we show that in every $r$-connected graph with $r\geq 2$, any two longest cycles meet in at least $Ω(r^{5/8})$ vertices, improving on $Ω(r^{3/5})$ from [Chen, Faudree and Gould, \emph{J. Combin. Theory, Ser.~ B}, 1998]. Our proof combines combinatorial arguments, computer-search and linear programming.

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arXiv2025

Extending Thomassen's conjecture to directed graphs

A famous conjecture by Thomassen from 1983 asserts that for any given $k,g\in \mathbb{N}$ there exists some $d=d(k,g)\in \mathbb{N}$ such that every graph of minimum degree at least $d$ contains a subgraph of minimum degree at least $k$ and girth at least $g$. In this paper, we initiate the systematic study of the directed analogs of Thomassen's conjecture one obtains when replacing minimum degree by minimum out-degree. Concretely, we study which digraphs $F$ are avoidable in the sense that there exists $d_F:\mathbb{N}\rightarrow \mathbb{N}$ such that every digraph of minimum out-degree at least $d_F(k)$ contains an $F$-free subdigraph of minimum out-degree at least $k$. Among our main results, we show that all orientations of $C_3$ and $C_5$ are avoidable, while one-directed orientations of complete bipartite graphs and all oriented trees are not avoidable. This, in particular, shows that the most direct extension of Thomassen's conjecture to digraphs is false. We also fully characterize which digraphs are avoidable when restricting the setting to regular host digraphs. Finally, we raise numerous attractive open problems in the hope of sparking further progress.

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arXiv2025

Short circuit walks in fixed dimension

Circuit augmentation schemes are a family of combinatorial algorithms for linear programming that generalize the simplex method. To solve the linear program, they construct a so-called monotone circuit walk: They start at an initial vertex of the feasible region and traverse a discrete sequence of points on the boundary, while moving along certain allowed directions (circuits) and improving the objective function at each step until reaching an optimum. Since the existence of short circuit walks has been conjectured (Circuit Diameter Conjecture), several works have investigated how well one can efficiently approximate shortest monotone circuit walks towards an optimum. A first result addressing this question was given by De Loera, Kafer, and Sanità [SIAM J. Opt., 2022], who showed that given as input an LP and the starting vertex, finding a $2$-approximation for this problem is NP-hard. Cardinal and the third author [Math. Prog. 2023] gave a stronger lower bound assuming the exponential time hypothesis, showing that even an approximation factor of $O(\frac{\log m}{\log \log m})$ is intractable for LPs defined by $m$ inequalities. Both of these results were based on reductions from highly degenerate polytopes in combinatorial optimization with high dimension. In this paper, we significantly strengthen the aforementioned hardness results by showing that for every fixed $\varepsilon>0$ approximating the problem on polygons with $m$ edges to within a factor of $O(m^{1-\varepsilon})$ is NP-hard. This result is essentially best-possible, as it cannot be improved beyond $o(m)$. In particular, this implies hardness for simple polytopes and in fixed dimension.

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arXiv2025

Critical edge sets in vertex-critical graphs

Criticality is a fundamental notion in graph theory that has been studied continually since its introduction in the early 50s by Dirac. A graph is called $k$-vertex-critical ($k$-edge-critical) if it is $k$-chromatic but removing any vertex (edge) lowers the chromatic number to $k-1$. A set of edges in a graph is called critical if its removal reduces the chromatic number of the graph. In 1970, Dirac conjectured a rather strong distinction between the notions of vertex- and edge-criticality, namely that for every $k\ge 4$ there exists a $k$-vertex-critical graph that does not have any critical edges. This conjecture was proved for $k\ge 5$ by Jensen in 2002 and remains open only for $k=4$. A much stronger version of Dirac's conjecture was proposed by Erdős in 1985: Let $k\ge 4$ be fixed, and let $f_k(n)$ denote the largest integer such that there exists a $k$-vertex-critical graph of order $n$ in which no set of at most $f_k(n)$ edges is critical. Is it true that $f_k(n)\rightarrow \infty$ for $n\rightarrow \infty$? Strengthening previous partial results, we solve this problem affirmatively for all $k>4$, proving that $$f_k(n)=Ω(n^{1/3}).$$ This leaves only the case $k=4$ open. We also show that a stronger lower bound of order $\sqrt{n}$ holds along an infinite sequence of numbers $n$. Finally, we provide a first non-trivial upper bound on the functions $f_k$ by proving that $$f_k(n)=O\left(\frac{n}{(\log n)^{Ω(1)}}\right)$$ for every $k\ge 4$. Our proof of the lower bound on $f_k(n)$ involves an intricate analysis of the structure of proper colorings of a modification of an earlier construction due to Jensen, combined with a gluing operation that creates new vertex-critical graphs without small critical edge sets from given such graphs. The upper bound is obtained using a variant of Szemerédi's regularity lemma due to Conlon and Fox.

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arXiv2025

Small hitting sets for longest paths and cycles

Motivated by an old question of Gallai (1966) on the intersection of longest paths in a graph and the well-known conjectures of Lovász (1969) and Thomassen (1978) on the maximum length of paths and cycles in vertex-transitive graphs, we present improved bounds for the parameters $\mathrm{lpt}(G)$ and $\mathrm{lct}(G)$, defined as the minimum size of a set of vertices in a graph $G$ hitting all longest paths (cycles, respectively). First, we show that every connected graph $G$ on $n$ vertices satisfies $\mathrm{lpt}(G)\le \sqrt{8n}$, and $\mathrm{lct}(G)\le \sqrt{8n}$ if $G$ is additionally $2$-connected. This improves a sequence of earlier bounds for these problems, with the previous state of the art being $O(n^{2/3})$. Second, we show that every connected graph $G$ satisfies $\mathrm{lpt}(G)\le O(\ell^{5/9})$, where $\ell$ denotes the maximum length of a path in $G$. As an immediate application of this latter bound, we present further progress towards Lovász' and Thomassen's conjectures: We show that every connected vertex-transitive graph of order $n$ contains a cycle (and path) of length $Ω(n^{9/14})$. This improves the previous best bound of the form $Ω(n^{13/21})$. Interestingly, our proofs make use of several concepts and results from structural graph theory, such as a result of Robertson and Seymour (1990) on transactions in societies and Tutte's $2$-separator theorem.

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arXiv2025

Proof of the KAMAK tree conjecture

There are many intriguing questions in extremal graph theory that are well-understood in the undirected setting and yet remain elusive for digraphs. A natural instance of such a problem was recently studied by Hons, Klimošová, Kucheriya, Mikšaník, Tkadlec and Tyomkyn: What are the digraphs that have to appear as a subgraph in all digraphs of sufficiently large minimum out-degree? Hons et al. showed that all such digraphs must be oriented forests with a specific structure, and conjectured that vice-versa all oriented forests with this specific structure appear in any digraph of sufficiently large minimum out-degree. In this paper, we confirm their conjecture.

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arXiv2025

Defective coloring of blowups

Given a graph $G$ and an integer $d\ge 0$, its $d$-defective chromatic number $χ^d(G)$ is the smallest size of a partition of the vertices into parts inducing subgraphs with maximum degree at most $d$. Guo, Kang and Zwaneveld recently studied the relationship between the $d$-defective chromatic number of the $(d+1)$-fold (clique) blowup $G\boxtimes K_{d+1}$ of a graph $G$ and its ordinary chromatic number, and conjectured that $χ(G)=χ^d(G\boxtimes K_{d+1})$ for every graph $G$ and $d\ge 0$. In this note we disprove this conjecture by constructing graphs $G$ of arbitrarily large chromatic number such that $χ(G)\ge \frac{30}{29}χ^d(G\boxtimes K_{d+1})$ for infinitely many $d$. On the positive side, we show that the conjecture holds with a constant factor correction, namely $χ^d(G\boxtimes K_{d+1})\le χ(G)\le 2χ^d(G\boxtimes K_{d+1})$ for every graph $G$ and $d\ge 0$.

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arXiv2025

Geometric realizations of dichotomous ordinal graphs

A dichotomous ordinal graph consists of an undirected graph with a partition of the edges into short and long edges. A geometric realization of a dichotomous ordinal graph $G$ in a metric space $X$ is a drawing of $G$ in $X$ in which every long edge is strictly longer than every short edge. We call a graph $G$ pandichotomous in $X$ if $G$ admits a geometric realization in $X$ for every partition of its edge set into short and long edges. We exhibit a very close relationship between the degeneracy of a graph $G$ and its pandichotomic Euclidean or spherical dimension, that is, the smallest dimension $k$ such that $G$ is pandichotomous in $\mathbb{R}^k$ or the sphere $\mathbb{S}^k$, respectively. First, every $d$-degenerate graph is pandichotomous in $\mathbb{R}^{d}$ and $\mathbb{S}^{d-1}$ and these bounds are tight for the sphere and for $\mathbb{R}^2$ and almost tight for $\mathbb{R}^d$, for $d\ge 3$. Second, every $n$-vertex graph that is pandichotomous in $\mathbb{R}^k$ has at most $μkn$ edges, for some absolute constant $μ<7.23$. This shows that the pandichotomic Euclidean dimension of any graph is linearly tied to its degeneracy and in the special cases $k\in \{1,2\}$ resolves open problems posed by Alam, Kobourov, Pupyrev, and Toeniskoetter. Further, we characterize which complete bipartite graphs are pandichotomous in $\mathbb{R}^2$: These are exactly the $K_{m,n}$ with $m\le 3$ or $m=4$ and $n\le 6$. For general bipartite graphs, we can guarantee realizations in $\mathbb{R}^2$ if the short or the long subgraph is constrained: namely if the short subgraph is outerplanar or a subgraph of a rectangular grid, or if the long subgraph forms a caterpillar.

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arXiv2025

Complexity of polytope diameters via perfect matchings

The Circuit diameter of polytopes was introduced by Borgwardt, Finhold and Hemmecke as a fundamental tool for the study of circuit augmentation schemes for linear programming and for estimating combinatorial diameters. Determining the complexity of computing the circuit diameter of polytopes was posed as an open problem by Sanità as well as by Kafer, and was recently reiterated by Borgwardt, Grewe, Kafer, Lee and Sanità. In this paper, we solve this problem by showing that computing the circuit diameter of a polytope given in halfspace-description is strongly NP-hard. To prove this result, we show that computing the combinatorial diameter of the perfect matching polytope of a bipartite graph is NP-hard. This complements a result by Sanità (FOCS 2018) on the NP-hardness of computing the diameter of fractional matching polytopes and implies the new result that computing the diameter of a $\{0,1\}$-polytope is strongly NP-hard, which may be of independent interest. In our second main result, we give a precise graph-theoretic description of the monotone diameter of perfect matching polytopes and use this description to prove that computing the monotone (circuit) diameter of a given input polytope is strongly NP-hard as well.

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arXiv2024

Local Shearer bound

We prove the following local strengthening of Shearer's classic bound on the independence number of triangle-free graphs: For every triangle-free graph $G$ there exists a probability distribution on its independent sets such that every vertex $v$ of $G$ is contained in a random independent set drawn from the distribution with probability $(1-o(1))\frac{\ln d(v)}{d(v)}$. This resolves the main conjecture raised by Kelly and Postle (2018) about fractional coloring with local demands, which in turn confirms a conjecture by Cames van Batenburg et al. (2018) stating that every $n$-vertex triangle-free graph has fractional chromatic number at most $(\sqrt{2}+o(1))\sqrt{\frac{n}{\ln(n)}}$. Addressing another conjecture posed by Cames van Batenburg et al., we also establish an analogous upper bound in terms of the number of edges. To prove these results we establish a more general technical theorem that works in a weighted setting. As a further application of this more general result, we obtain a new spectral upper bound on the fractional chromatic number of triangle-free graphs: We show that every triangle-free graph $G$ satisfies $χ_f(G)\le (1+o(1))\frac{ρ(G)}{\ln ρ(G)}$ where $ρ(G)$ denotes the spectral radius. This improves the bound implied by Wilf's classic spectral estimate for the chromatic number by a $\ln ρ(G)$ factor and makes progress towards a conjecture of Harris on fractional coloring of degenerate graphs.

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arXiv2024

Fractional chromatic number vs. Hall ratio

Given a graph $G$, its Hall ratio $ρ(G)=\max_{H\subseteq G}\frac{|V(H)|}{α(H)}$ forms a natural lower bound on its fractional chromatic number $χ_f(G)$. A recent line of research studied the fundamental question of whether $χ_f(G)$ can be bounded in terms of a (linear) function of $ρ(G)$. In a breakthrough-result, Dvořák, Ossona de Mendez and Wu gave a strong negative answer by proving the existence of graphs with bounded Hall ratio and arbitrarily large fractional chromatic number. In this paper, we solve two natural follow-up problems that were raised by Dvořák et al. The first problem concerns determining the growth of $g(n)$, defined as the maximum ratio $\frac{χ_f(G)}{ρ(G)}$ among all $n$-vertex graphs. Dvořák et al. obtained the bounds $Ω(\log\log n) \le g(n)\le O(\log n)$, leaving an exponential gap between the lower and upper bound. We almost fully resolve this problem by proving that the truth is close to the upper bound, i.e., $g(n)=(\log n)^{1-o(1)}$. The second problem posed by Dvořák et al. asks for the existence of graphs with bounded Hall ratio, arbitrarily large fractional chromatic number and such that every subgraph contains an independent set that touches a constant fraction of its edges. We affirmatively solve this second problem by showing that such graphs indeed exist.

math.CO↗

arXiv2024

Clustered Colouring of Odd-$H$-Minor-Free Graphs

The clustered chromatic number of a graph class $\mathcal{G}$ is the minimum integer $c$ such that every graph $G\in\mathcal{G}$ has a $c$-colouring where each monochromatic component in $G$ has bounded size. We study the clustered chromatic number of graph classes $\mathcal{G}_H^{\text{odd}}$ defined by excluding a graph $H$ as an odd-minor. How does the structure of $H$ relate to the clustered chromatic number of $\mathcal{G}_H^{\text{odd}}$? We adapt a proof method of Norin, Scott, Seymour and Wood (2019) to show that the clustered chromatic number of $\mathcal{G}_H^{\text{odd}}$ is tied to the tree-depth of $H$.

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arXiv2024

Chromatic number and regular subgraphs

In 1992, Erdős and Hajnal posed the following natural problem: Does there exist, for every $r\in \mathbb{N}$, an integer $F(r)$ such that every graph with chromatic number at least $F(r)$ contains $r$ edge-disjoint cycles on the same vertex set? We solve this problem in a strong form, by showing that there exist $n$-vertex graphs with fractional chromatic number $Ω\left(\frac{\log \log n}{\log \log \log n}\right)$ that do not even contain a $4$-regular subgraph. This implies that no such number $F(r)$ exists for $r\ge 2$. We show that assuming a conjecture of Harris, the bound on the fractional chromatic number in our result cannot be improved.

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arXiv2024

Resolution of the Kohayakawa-Kreuter conjecture

A graph $G$ is said to be Ramsey for a tuple of graphs $(H_1,\dots,H_r)$ if every $r$-coloring of the edges of $G$ contains a monochromatic copy of $H_i$ in color $i$, for some $i$. A fundamental question at the intersection of Ramsey theory and the theory of random graphs is to determine the threshold at which the binomial random graph $G_{n,p}$ becomes a.a.s. Ramsey for a fixed tuple $(H_1,\dots,H_r)$, and a famous conjecture of Kohayakawa and Kreuter predicts this threshold. Earlier work of Mousset-Nenadov-Samotij, Bowtell-Hancock-Hyde, and Kuperwasser-Samotij-Wigderson has reduced this probabilistic problem to a deterministic graph decomposition conjecture. In this paper, we resolve this deterministic problem, thus proving the Kohayakawa-Kreuter conjecture. Along the way, we prove a number of novel graph decomposition results which may be of independent interest.

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arXiv2024

On the difference between the chromatic and cochromatic number

The cochromatic number $ζ(G)$ of a graph $G$ is the smallest number of colors in a vertex-coloring of $G$ such that every color class forms an independent set or a clique. In three papers written around 1990, Erdős, Gimbel and collaborators raised several open problems regarding the relationship of the chromatic and cochromatic number of a graph. In this short note, we address several of these problems, in particular -we disprove a conjecture of Erdős, Gimbel and Straight from 1988, -answer negatively a problem posed by Erdős and Gimbel in 1993, and -give positive evidence for a 1000\$--question of Erdős and Gimbel.

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arXiv2024

Topological Drawings meet Classical Theorems from Convex Geometry

In this article we discuss classical theorems from Convex Geometry in the context of topological drawings and beyond. In a simple topological drawing of the complete graph $K_n$, any two edges share at most one point: either a common vertex or a point where they cross. Triangles of simple topological drawings can be viewed as convex sets. This gives a link to convex geometry. As our main result, we present a generalization of Kirchberger's Theorem that is of purely combinatorial nature. It turned out that this classical theorem also applies to "generalized signotopes" - a combinatorial generalization of simple topological drawings, which we introduce and investigate in the course of this article. As indicated by the name they are a generalization of signotopes, a structure studied in the context of encodings for arrangements of pseudolines. We also present a family of simple topological drawings with arbitrarily large Helly number, and a new proof of a topological generalization of Carathéodory's Theorem in the plane and discuss further classical theorems from Convex Geometry in the context of simple topological drawings.

math.CO↗

arXiv2024

Topological Minors in Typical Lifts

An $\ell$-lift of a graph $G$ is any graph obtained by replacing every vertex of $G$ with an independent set of size $\ell$, and connecting every pair of two such independent sets that correspond to an edge in $G$ by a matching of size $\ell$. Graph lifts have found numerous interesting applications and connections to a variety of areas over the years. Of particular importance is the random graph model obtained by considering an $\ell$-lift of a graph sampled uniformly at random. This model was first introduced by Amit and Linial in 1999, and has been extensively investigated since. In this paper, we study the size of the largest topological clique in random lifts of complete graphs. In 2006, Drier and Linial raised the conjecture that almost all $\ell$-lifts of the complete graph on $n$ vertices contain a subdivision of a clique of order $Ω(n)$ as a subgraph provided $\ell$ is at least linear in $n$. We confirm their conjecture in a strong form by showing that for $\ell \ge (1+o(1))n$, one can almost surely find a subdivision of a clique of order $n$. We prove that this is tight by showing that for $\ell \le (1-o(1))n$, almost all $\ell$-lifts do not contain subdivisions of cliques of order $n$. Finally, for $2 \le \ell \ll n$, we show that almost all $\ell$-lifts of $K_n$ contain a subdivision of a clique on $(1-o(1))\sqrt{\frac{2n \ell}{1-1/\ell}}$ vertices and that this is tight up to the lower order term.

math.CO↗

arXiv2024

Optimal bounds for zero-sum cycles. I. Odd order

For a finite (not necessarily Abelian) group $(Γ,\cdot)$, let $n(Γ) \in \mathbb{N}$ denote the smallest positive integer $n$ such that for every labelling of the arcs of the complete digraph of order $n$ using elements from $Γ$, there exists a directed cycle such that the arc-labels along the cycle multiply to the identity. Alon and Krivelevich initiated the study of the parameter $n(\cdot)$ on cyclic groups and proved $n(\mathbb{Z}_q)=O(q \log q)$. This was later improved to a linear bound of $n(Γ)\le 8|Γ|$ for every finite Abelian group by Mészáros and the last author, and then further to $n(Γ)\le 2|Γ|-1$ for every non-trivial finite group independently by Berendsohn, Boyadzhiyska and Kozma as well as by Akrami, Alon, Chaudhury, Garg, Mehlhorn and Mehta. In this series of two papers we conclude this line of research by proving that $n(Γ)\le |Γ|+1$ for every finite group $(Γ,\cdot)$, which is the best possible such bound in terms of the group order and precisely determines the value of $n(Γ)$ for all cyclic groups as $n(\mathbb{Z}_q)=q+1$. In the present paper we prove the above result for all groups of odd order. The proof for groups of even order needs to overcome substantial additional obstacles and will be presented in the second part of this series.

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