arXiv subjects

Raphael Steiner

Publications and source records attributed to Raphael Steiner.

At least 19 recordsLinked to original sources

arXiv2026

Proof of Lichiardopol's conjecture on disjoint directed cycles of distinct lengths

There is a fascinating array of interrelated questions studying which structures can be guaranteed in digraphs of large minimum out-degree. These often have intriguingly simple statements, yet seem surprisingly difficult to approach. A well-known example is Lichiardopol's conjecture (2014), stating that there exists a function $g:\mathbb{N}\rightarrow \mathbb{N}$ such that every digraph with minimum out-degree at least $g(k)$ contains $k$ vertex-disjoint directed cycles of distinct lengths. In this paper, building on earlier work of the second author, we confirm this conjecture in full generality. We also generalise this result to a weighted setting. Our proof uses and combines many ingredients from structural digraph theory such as butterfly minors, directed tangles, a directed analogue of the Tangle-Wall Theorem due to Robertson and Seymour as well as a local variant of the Directed Flat Wall Theorem due to Giannopoulou, Kawarabayashi, Kreutzer and Kwon. These techniques, which are somewhat atypical in the study of minimum degree conditions, may be of independent interest and may find further applications.

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arXiv2026

A relaxation of the Bermond-Thomassen conjecture

The well-known Bermond-Thomassen conjecture states that every digraph of minimum out-degree at least $2k-1$ contains $k$ vertex-disjoint directed cycles. Despite being posed in 1981, this conjecture remains unresolved for all $k \ge 4$. We prove a relaxation of this conjecture: every digraph $D$ of minimum out-degree at least $2k-1$ contains $k$ vertex-disjoint cycles, each of which either is directed or can be made directed by reversing one of its arcs. This bound is sharp and answers a question raised by Cames van Batenburg during the online workshop "Entropy Compression and Related Methods" in $2021$.

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arXiv2026

Locally bipartite subgraphs via multicolor Ramsey numbers

A famous conjecture of Erd\H{o}s and Hajnal (1969) states that for every integer $g\ge 4$ there is a smallest function $f_g:\mathbb{N}\to\mathbb{N}$ such that every graph of chromatic number at least $f_g(k)$ contains a subgraph of chromatic number $k$ and girth at least $g$. So far, this has only been proved for $g=4$ by R\"odl (1977), with $f_4(k)$ bounded by a tower of height $\Theta(k^2\log k)$. We exhibit a surprising connection between finding high-chromatic subgraphs of large odd-girth (avoiding short odd cycles) and lower-bounding multicolor Ramsey numbers of odd cycles. Using this connection, we prove that for every odd $g\ge 5$ there is a function $h_g:\mathbb{N}\to\mathbb{N}$ growing as a power tower of height $\frac{g-3}{2}$ such that every graph of chromatic number at least $h_g(k)$ contains a subgraph of chromatic number at least $k$ and odd-girth at least $g$. This proves a conjecture of Mohar and Wu (2018), addresses a question of Erd\H{o}s and Hajnal (1975), and for $g=5$ improves R\"odl's bound on $f_4(k)$ to a single-exponential. We extend this to a much more general meta-theorem which applies to many graph parameters: if $f$ is the fractional chromatic number, the Hall ratio, or the strict vector chromatic number (Lov\'{a}sz-Theta-function of the complement), then for every $k,g\in\mathbb{N}$, every graph $G$ with sufficiently large $f(G)$ contains a subgraph $G'$ of odd-girth at least $g$ with $f(G')\ge k$. The key Ramsey-theoretic ingredient is a new lower bound on Ramsey numbers of odd cycles. For $p\ge 1$, let $\mathcal{O}_p=\{C_3,C_5,\ldots,C_{2p+1}\}$. We show that $R_k(\mathcal{O}_p)\ge(\log^{(p-1)}k)^{k/3-o(k)}$ for every fixed $p$, where $\log^{(p-1)}$ denotes the $(p-1)$-fold iterated logarithm. This yields the first superexponential lower bound on multicolor Ramsey numbers of fixed odd cycles, and extends the recent breakthrough by OpenAI for triangles.

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arXiv2026

Multicolor Ramsey numbers of odd cycles are superexponential

In a recent breakthrough, OpenAI proved that the $k$-color Ramsey number of the triangle $C_3$ grows super-exponentially, more precisely, they proved that $R_k(C_3)\ge k^{k/3-o(k)}$. In this short note, we present a modification of their recursive construction that works for multicolor Ramsey numbers of fixed odd cycles. More precisely, for $p\ge 1$, let $\mathcal{O}_p=\{C_3,C_5,\ldots,C_{2p+1}\}$. We show that \[ R_k(\mathcal{O}_p)\ge (\log^{(p-1)}k)^{k/3-o(k)} \] for every fixed $p$, where $\log^{(p-1)}$ denotes the $(p-1)$-fold iterated logarithm. This immediately implies that for every fixed odd cycle, the multicolor Ramsey number is superexponential in the number of colors. The presented proof was found autonomously by ChatGPT 5.6 Pro/Sol.

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arXiv2026

Disproof of the tree product conjecture via the Heisenberg group

Product structure theory aims to understand complex graphs by embedding them into products of simpler graphs. In this direction, Campbell, Distel, Gollin, Harvey, Hendrey, Hickingbotham, Mohar and Wood (2022) put forth the conjecture that all graphs of degree-$d$ polynomial growth (i.e., where balls of radius $r$ have $\mathcal{O}(r^d)$ vertices) can be embedded into the strong product of $d$ trees, each with linear growth, and a constant-size clique. In this paper, we disprove this conjecture for $d = 4$. The counterexamples are finite subgraphs of a Cayley graph of the discrete $3$-dimensional Heisenberg group $\mathbb{H}(\mathbb{Z})$. These graphs were first proposed by Huang and McCarty as potential counterexamples to the conjecture. A key technical tool of our proof is the ''quantitative central collapse'' theorem due to Cheeger, Kleiner and Naor (2011), guaranteeing that every Lipschitz map from the continuous Heisenberg group $\mathbb{H}$ to the function space $L_1$ collapses along a central line.

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arXiv2026

A constant-factor step towards Vizing's conjecture

Vizing's conjecture from 1963, considered by many the most important open problem in the field of graph domination, states that all graphs $G$ and $H$ satisfy $$\gamma(G\square H)\ge \gamma(G)\gamma(H),$$ where $\gamma$ denotes the domination number and $\square$ the Cartesian product. In a seminal result, Clark and Suen (2000) proved an approximate form of the conjecture, namely that $\gamma(G\square H)\ge \frac{1}{2}\gamma(G)\gamma(H)$ for all graphs $G$ and $H$. Despite several lower-order improvements of this bound and improvements for special classes of graphs $G$ and $H$, no absolute constant $c>\frac{1}{2}$ such that $\gamma(G\square H)\ge c\gamma(G)\gamma(H)$ for all graphs $G$ and $H$, has been known thus far. In this paper, we obtain the first constant-factor improvement of the Clark-Suen bound by proving that for all graphs $G$ and $H$, we have $$\gamma(G\square H)\ge c\gamma(G)\gamma(H),$$ where $$c=\frac{5+\sqrt{73}}{24}\approx 0.5643.$$ Along the way, we prove another lower bound on $\gamma(G\square H)$ which outperforms the above bound for many graphs.

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arXiv2026

Towards the Lov\'{a}sz conjecture via sublinear expanders

Lov\'{a}sz' famous Hamiltonicity conjecture (1969) states that every connected vertex-transitive graph has a Hamiltonian path. A stronger version of the conjecture, often attributed to Thomassen (1978), states that every sufficiently large such graph even has a Hamiltonian cycle. Despite the great amount of attention these conjectures have attracted over the past decades both in the combinatorial and algebraic communities, for more than 40 years the best known lower bound for the maximum length of a cycle (path) in a connected vertex-transitive graph of order $n$ remained of the form $\Omega(\sqrt{n})$, due to Babai (1979). A series of recent works has successively improved the exponent in this lower bound further. In this paper, improving the previous state-of-the-art bound $\Omega(n^{9/14})$ due to Norin et al.~(2025), we prove that every connected vertex-transitive graph of order $n$ contains a cycle of length at least $n^{2/3-o(1)}$. This hits a natural barrier for several existing approaches from previous work. Our proofs combine recent embedding techniques for paths in sublinear expanders, sublinear expander decompositions of almost-regular graphs, and several additional combinatorial ideas.

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arXiv2026

The Dominating 4-Colour Theorem

A "dominating $K_t$-model" in a graph $G$ is a sequence $(T_1,\dots,T_t)$ of pairwise vertex-disjoint connected subgraphs of $G$, such that whenever $1\leq i<j\leq t$ every vertex in $T_j$ has a neighbour in $T_i$. Replacing "every vertex in $T_j$" by "some vertex in $T_j$" retrieves the standard definition of $K_t$-model, which is equivalent to a $K_t$-minor in $G$. We prove that every graph with no dominating $K_5$-model is $4$-colourable. This generalises and is significantly stronger than the 4-colour theorem for planar graphs or for graphs with no $K_5$-minor. It also makes progress towards Haj\'{o}s' conjecture on $K_5$-subdivisions in $5$-chromatic graphs.

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arXiv2026

Openly disjoint cycles and directed tree-width of regular digraphs

Given a digraph $D$, let $c(D)$ denote the largest integer $k$ such that there are $k$ openly disjoint cycles through a vertex, i.e., a collection of directed cycles $C_1,\ldots,C_k$ through a common vertex $v$ such that $C_1-v,\ldots,C_k-v$ are pairwise vertex-disjoint. The famous Caccetta-H\"aggkvist conjecture and its regular variant due to Behzad, Chartrand and Wall from 1970, have motivated the study of degree conditions forcing $c(D)$ to be large. In 1985 Thomassen constructed digraphs of arbitrarily high minimum out- and in-degree such that $c(D)\le 2$. In 2005, Seymour asked whether in contrast every $r$-regular digraph satisfies $c(D)=r$, which would have implied the Behzad-Chartrand-Wall conjecture. In 2008, Mader answered this negatively for every $r\ge 8$, but conjectured that nevertheless the minimum value $c_r$ of $c(D)$ over all $r$-regular digraphs grows with $r$, i.e. $\lim_{r\rightarrow\infty}c_r=\infty$. As the first main result of our paper, we prove Mader's conjecture in a strong form by showing $c_r\ge \lceil\frac{3}{22} r\rceil$ for every $r\in \mathbb{N}$. We also show $c_r\le 7\left\lceil \frac{r}{8}\right\rceil$, improving the previous best upper bound $c_r\le r-\Theta(\sqrt{r})$ due to Mader. In our second main result we show that every $r$-regular digraph has directed tree-width $\Omega(r)$. This is tight up to the implied constant and cannot be extended to digraphs of minimum out- and in-degree at least $r$. As a corollary we obtain the existence of a function $f:\mathbb{N}\rightarrow \mathbb{N}$ such that every regular digraph with degree at least $f(k)$ contains a subdivision of the cylindrical wall of order $k$, and hence of a large class of planar digraphs. This makes progress on the notoriously difficult problem of finding degree conditions guaranteeing subdivisions of digraphs, related to a well-known conjecture of Mader from 1985.

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arXiv2026

A note on Ramsey numbers for minors

Let $R_h(k; \ell)$ be the smallest integer $n$ such that any edge coloring of a complete graph on $n$ vertices in $\ell$ colors results in a monochromatic $K_k$-minor, in other words, a graph with Hadwiger number $k$, i.e., a graph that could be transformed into a clique $K_k$ on $k$ vertices via a sequence of edge contractions and vertex deletions. More generally, for a graph $F$ and integer $\ell$ let $R_h(F;\ell)$ be the smallest integer $n$ such that any edge coloring of a complete graph on $n$ vertices in $\ell$ colors results in a monochromatic $F$-minor. In 2001 Thomason and in 2005 Myers and Thomason asymptotically determined the extremal numbers for clique minors and $F$-minors, respectively. They found the respective explicitly computable leading constants $\beta=0.265656...$ and $\gamma(F)\cdot \beta$ for these extremal numbers. We determine $R_h(F;2)$ for every graph $F$ as $$R_h(F;2)=(\gamma(F)+o(1))|V(F)|\sqrt{\log_2(|V(F)|)},$$ where the $o(1)$-term tends to zero as $|V(F)|\rightarrow \infty$. In particular, $$R_h(k;2)=(1+o(1))k\sqrt{\log_2 k}.$$ When $\ell\gg k \gg 1$, we show that $$ R_h(k; \ell) = (2\beta+o(1)) \ell k \sqrt{\log_2 k}.$$

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arXiv2026

Finding Short Paths on Simple Polytopes

We prove that computing a shortest monotone path to the optimum of a linear program over a simple polytope is NP-hard, thus resolving a 2022 open question of De Loera, Kafer, and Sanit\`a. As a consequence, finding a shortest sequence of pivots to an optimal basis with the simplex method is NP-hard. In fact, we show this is NP-hard already for fractional knapsack polytopes. By applying an additional polyhedral construction, we show that computing the diameter of a simple polytope is NP-hard, resolving a 2003 open problem by Kaibel and Pfetsch. Finally, on the positive side we show that every polytope has a small, simple extended formulation for which a linear length path may be found between any pair of vertices in polynomial time building upon a result of Kaibel and Kukharenko.

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arXiv2026

Long cycles in vertex transitive digraphs

One of the most well-known conjectures concerning Hamiltonicity in graphs asserts that any sufficiently large connected vertex transitive graph contains a Hamilton cycle. In this form, it was first written down by Thomassen in 1978, inspired by a closely related conjecture due to Lov\'asz from 1969. It has been attributed to several other authors in a survey on the topic by Witte and Gallian in 1984. The analogous question for vertex transitive digraphs has an even longer history, having been first considered by Rankin in 1946. It is arguably more natural from the group-theoretic perspective underlying this problem in both settings. Trotter and Erd\H{o}s proved in 1978 that there are infinitely many connected vertex transitive digraphs which are not Hamiltonian. This left open the very natural question of how long a directed cycle one can guarantee in a connected vertex transitive digraph on $n$ vertices. In 1981, Alspach asked if the maximum perimeter gap (the gap between the circumference and the order of the digraph) is a growing function in $n$. We answer this question in the affirmative, showing that it grows at least as fast as $(1-o(1)) \ln n$. On the other hand, we prove that one can always find a directed cycle of length at least $\Omega(n^{1/3})$, establishing the first lower bound growing with $n$, providing a directed analogue of a famous result of Babai from 1979 in the undirected setting.

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arXiv2026

Coloring small locally sparse degenerate graphs and related problems

The classic upper bound on the chromatic number of $d$-degenerate graphs is $d+1$, shown to be tight by complete graphs. A natural question is whether this bound remains tight if one forbids large cliques. Classic constructions of Tutte and Zykov from the early 50s show that there exist $d$-degenerate $(d+1)$-chromatic graphs that are triangle-free, however these constructions grow rapidly with $d$. Motivated by this and addressing a problem posed by the second author at the Oberwolfach Graph Theory workshop, we prove that the minimum order $f(d)$ of a $d$-degenerate triangle-free graph of chromatic number $d+1$ satisfies $e^{\Omega(d)}\le f(d)\le e^{O(d^2\log d)}.$ The lower bound follows from a novel upper bound on the chromatic number of triangle-free graphs: Every triangle-free $d$-degenerate graph $G$ on $n \le e^{O(d)}$ vertices satisfies $$\chi(G)\le O\left(\frac{d}{\log\left(d/\log n\right)}\right).$$ We extend this to a more general result about degenerate graphs with sparse neighborhoods, which has applications to many graph coloring problems: For example, we prove that every counterexample to Hadwiger's conjecture with parameter $t$ must have a complete bipartite subgraph with one exponentially large side ($K_{a,b}$ where $a=(\log t)^{1/2-o(1)}$ and $b=e^{t^{1-o(1)}}$) or a small and very dense subgraph (of order $\le t$ with $t^{2-o(1)}$ edges) in some neighborhood. For the upper bound on $f(d)$ we establish a surprising connection between $f(d)$ and the on-line-chromatic number $g(n)$ of $n$-vertex triangle-free graphs. We also give an asymptotic improvement of the previous best upper bound for $g(n)$ due to Lov\'{a}sz, Saks and Trotter from 1989. Along the way we disprove a generalization of Harris' fractional coloring conjecture to graphs of bounded clique number and raise numerous problems which open up interesting directions to explore for future research.

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arXiv2025

Disproof of the Odd Hadwiger Conjecture

We prove that there exist graphs which do not contain $K_t$ as an odd minor and whose chromatic number is at least $(\frac 32-o(1))t$. This disproves, in a strong form, the odd Hadwiger conjecture of Gerards and Seymour from 1993.

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arXiv2025

Nowhere-zero flow reconfiguration

We initiate the study of nowhere-zero flow reconfiguration. The natural question is whether any two nowhere-zero $k$-flows of a given graph $G$ are connected by a sequence of nowhere-zero $k$-flows of $G$, such that any two consecutive flows in the sequence differ only on a cycle of $G$. We study this problem in the setting of integer flows and group flows, and prove a number of positive and negative results. * The natural reconfiguration variant of Tutte's 5-flow conjecture, stating that any two nowhere-zero 5-flows in any 2-edge-connected graph are connected, is false in the group and integer cases. * All nowhere-zero $\mathbb{Z}_2^8$-flows of every 2-edge-connected graph are connected and for every sufficiently large abelian group $A$, all nowhere-zero $A$-flows of every 2-edge-connected graph are connected. * The group structure affects the answer, contrary to the existence problem for nowhere-zero flows. * We highlight a duality with recoloring in planar graphs and deduce that any two nowhere-zero 7-flows in a planar graph are connected, among other results. * For every 2-edge-connected graph $G$, there is an integer $k$ such that all nowhere-zero $k$-flows of $G$ are connected.

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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 Sanit\`a [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 Erd\H{o}s 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 $$f_k(n)=\Omega(n^{1/3}).$$ 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 $$f_k(n)=O\left(\frac{n}{(\log n)^{\Omega(1)}}\right)$$ 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 Szemer\'{e}di's regularity lemma due to Conlon and Fox.

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