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James Davies

Publications and source records attributed to James Davies.

At least 19 recordsLinked to original sources
arXiv2026

Revisiting separators in string graphs, region intersection graphs, and induced-minor-free graphs

String graphs are intersection graphs of continuous curves in the plane. In 2010, Fox and Pach proved that string graphs with $m$ edges admit balanced vertex separators of size $\mathcal{O}(m^{3/4}\sqrt{\log m})$ and conjectured that this bound can be improved to $\mathcal{O}(\sqrt{m})$, which would be optimal. After partial progress, Lee claimed in 2017 to prove this conjecture, and more generally to prove analogous separator theorems for region intersection graphs over minor-closed graph families. A key intermediate step in Lee's approach was a padded decomposition scheme for region intersection graphs. In 2024, Korhonen and Lokshtanov extended Lee's approach to the broader setting of graph classes excluding a fixed graph as an induced minor. These results have since found applications in a wide range of works. Later in 2024, a flaw was discovered in the construction of Lee's padded decomposition scheme, invalidating Lee's as well as Korhonen and Lokshtanov's separator results, and many works relying on them. In this work, we prove that string graphs with $m$ edges admit balanced separators of size $\mathcal{O}(\sqrt{m})$, and more generally obtain analogous $\mathcal{O}(\sqrt{m})$ bounds for region intersection graphs over fixed minor-closed graph families and for graphs excluding a fixed graph as an induced minor, thereby repairing the vast majority of the affected works. Furthermore, our balanced separators can be found in polynomial time. Our proof uses a different approach from Lee's and does not rely on padded decompositions. Since padded decompositions have also found applications independent of the separator theorems, particularly for string graphs, we also prove a padded decomposition result for region intersection graphs that is optimal for string graphs. We have formalized both the balanced separator and the padded decomposition results in Lean 4 with mathlib.

math.CO↗
arXiv2026

Burling graphs in graphs with large chromatic number

A graph class is $χ$-bounded if the only way to force large chromatic number in graphs from the class is by forming a large clique. In the 1970s, Erdős conjectured that intersection graphs of straight-line segments in the plane are $χ$-bounded, but this was disproved by Pawlik et al. (2014), who showed another way to force large chromatic number in this class -- by triangle-free graphs $B_k$ with $χ(B_k)=k$ constructed by Burling (1965). This also disproved the celebrated conjecture of Scott (1997) that classes of graphs excluding induced subdivisions of a fixed graph are $χ$-bounded. We prove that in broad classes of graphs excluding induced subdivisions of a fixed graph, including the increasingly more general classes of segment intersection graphs, string graphs, region intersection graphs, and hereditary classes of graphs with finite asymptotic dimension, large chromatic number can be forced only by large cliques or large graphs $B_k$. One corollary is that the hereditary closure of $\{B_k\colon k\geq 1\}$ forms a minimal hereditary graph class with unbounded chromatic number -- the second known graph class with this property after the class of complete graphs. Another corollary is that the decision variant of approximate coloring in the aforementioned graph classes can be solved in polynomial time by exhaustively searching for a sufficiently large clique or copy of $B_k$. We also discuss how our results along with some results of Chudnovsky, Scott, and Seymour on the existence of colorings can be turned into polynomial-time algorithms for the search variant of approximate coloring in string graphs (with intersection model in the input) and other aforementioned graph classes. Such an algorithm has not yet been known for any graph class that is not $χ$-bounded.

math.CO↗
arXiv2026

Measurable obstructions for unmeasurable colourings

Due to the availability of powerful analytic techniques, vastly superior lower bounds are known for the measurable chromatic number of Euclidean spaces compared to their ordinary chromatic number. Indeed, even the breakthrough lower bound of 5 for the famous Hadwiger-Nelson problem lagged over 35 years behind that of the measurable setting. This raises the fundamental question of whether the measurable and ordinary chromatic number of Euclidean spaces differ as conjectured by Székely in 1984. Our main result is that $\overlineα(\mathbb{R}^4)=m_1(\mathbb{R}^4)$ and $χ(\mathbb{R}^4)=χ^{(m)}(\mathbb{R}^4)$, and for $d\ge5$ that \[ \overlineα(\mathbb{Q}^d) = \overlineα(\mathbb{R}^d)=m_1(\mathbb{R}^d) \qquad\text{and}\qquad χ(\mathbb{Q}^d) = χ(\mathbb{R}^d)=χ^{(m)}(\mathbb{R}^d). \] Our theorem also holds for multiple forbidden distances $D=\{d_1,\ldots,d_t\}$ provided that $d_1^2,\ldots,d_t^2 \in \mathbb{Q}$. As a consequence, we immediately lift numerous measurable chromatic number results into the ordinary setting. We also take the opportunity to further optimize the new bounds. For multiple distances, Erdős asked whether the chromatic number of $\mathbb{R}^d$ with up to $k$ forbidden distances $D$ grows exponentially in $k$. By a theorem of Bukh, we obtain for $d \ge 4$ that \[ \sup_{|D|=k}χ_D(\mathbb{R}^d) \ge m_1(\mathbb{R}^d)^{-k}. \] Making progress on another problem of Erdős, we prove that \[ (2+o(1))^d \le χ(\mathbb{R}^d) \le \left(\frac{3\sqrt{3}}{4}+o(1)\right)^d. \] We also vastly improve the lower bounds for $χ(\mathbb{R}^d)$ for small $d\ge4$. We expect that our techniques could be developed much further. This includes the possibility of extending our main theorem that $χ(\mathbb{R}^d)=χ^{(m)}(\mathbb{R}^d)$ for $d\ge 4$ to $d=3$ or possibly even to $d=2$ to tackle the Hadwiger-Nelson problem.

math.CO↗
arXiv2026

Hadwiger--Nelson Problem for Typical Norms

The classical Hadwiger-Nelson problem asks for the chromatic number of the unit distance graph of the Euclidean plane. Over the years, this problem has been considered for a variety of other normed spaces, with higher-dimensional Euclidean space $\mathbb{R}^d$ being perhaps the most natural and well-studied. Alon, Bucić, and Sauermann proved that, for a typical norm on $\mathbb{R}^d$, the chromatic number of the unit distance graph is at most $2^d$. We improve this exponential bound to a linear one by showing that the chromatic number of a typical norm on $\mathbb{R}^d$ is at most $2d$ and that this is tight. This shows a stark difference in the behavior compared to the Euclidean case, where there is an exponential lower bound. One of the key ingredients is a certain high-dimensional, matrix generalization of the Lonely runner conjecture, which also allows us to completely settle the so-called view-obstruction conjecture of Schoenberg from 1978 and a more recent covering-radius conjecture of Henze and Malikiosis.

math.CO↗
arXiv2026

The coarse Erdős-Pósa theorem

We prove the coarse Erdős-Pósa conjecture of Georgakopoulos and Papasoglu. Informally, any graph either contains many fat cycles that are pairwise far apart, or there is a small number of bounded radius balls that together hit all of them. To be more precise, if $G$ is a graph with no $q$-fat model of $k \cdot K_3$ for some $q, k \in \mathbb{N}$, then there is a set $X\subseteq V(G)$ of $\mathcal{O}(k\log k)$ vertices such that every $q$-fat model of $K_3$ in $G$ has distance $\mathcal{O}(q)$ from $X$. In another form more closely resembling Manning's theorem that characterises quasi-trees: if $G$ is a graph with no $q$-fat model of $k \cdot K_3$ for some $q, k \in \mathbb{N}$, then $G$ is $\mathcal{O}(q)$-quasi-isometric to a graph $H$ that contains a set $X\subseteq V(H)$ of $\mathcal{O}(k\log k)$ vertices such that $H-X$ is a forest. Bootstrapping this result, we further prove that every graph with no $q$-fat model of $k \cdot K_3$ is quasi-isometric to a graph with no $k \cdot K_3$ minor, where the quasi-isometry can be chosen to only have additive distortion. This result also holds for $k = \infty$. We also obtain an Erdős-Pósa theorem for long induced cycles that are far apart.

math.CO↗
arXiv2026

CAT(0) square complexes that do not embed into finite products of trees

Answering a question of Chepoi and Hagen, we give two constructions of bounded degree CAT(0) square complexes that cannot be isometrically embedded into any finite product of trees. The first is based on Burling graphs and has degree at most five, which is optimal. These square complexes topologically embed into $\R^3$ since we further prove that every CAT(0) square complex whose vertices have degree at most five can be topologically embedded into the product of a line and a star. It turns out that in this setting, Burling graphs are the only obstruction: we prove that CAT(0) square complexes that topologically embed into $\R^3$ and whose crossing graph forbids some induced Burling graph can be isometrically embedded into a finite product of trees. Our second CAT(0) square complex has degree at most six and its crossing graph forbids an induced Burling graph. Of independent graph-theoretic interest, along the way we construct intersections of chordal and interval graphs with arbitrarily large girth and chromatic number. We also discuss connections to nice labellings of event structures.

math.MG↗
arXiv2026

On the chromatic number of the union of comparability graphs

Resolving in a strong sense a problem of Gyárfás on the union of two perfect graphs, we prove that for every pair of positive integers $d$ and $k$, there is a graph $G$ with clique number $k$ and chromatic number $k^d$ that is the union of $d$ comparability graphs. We also show that the chromatic number can be replaced by the fractional chromatic number or $\frac{|V(G)|}{α(G)}$.

math.CO↗
arXiv2026

String graphs are quasi-isometric to planar graphs

We prove that for every countable string graph $S$, there is a planar graph $G$ with $V(G)=V(S)$ such that \[ \frac{1}{23660800}d_S(u,v) \le d_G(u,v) \le 162 d_S(u,v) \] for all $u,v\in V(S)$, where $d_S(u,v)$, $d_G(u,v)$ denotes the distance between $u$ and $v$ in $S$ and $G$ respectively. In other words, string graphs are quasi-isometric to planar graphs. This theorem lifts a number of theorems from planar graphs to string graphs, we give some examples. String graphs have Assouad-Nagata (and asymptotic dimension) at most 2. Connected, locally finite, quasi-transitive string graphs are accessible. A finitely generated group $Γ$ is virtually a free product of free and surface groups if and only if $Γ$ is quasi-isometric to a string graph. Two further corollaries are that countable planar metric graphs and complete Riemannian planes are also quasi-isometric to planar graphs, which answers a question of Georgakopoulos and Papasoglu. For finite string graphs and planar metric graphs, our proofs yield polynomial time (for string graphs, this is in terms of the size of a representation given in the input) algorithms for generating such quasi-isometric planar graphs. We further extend our techniques to show that every complete Riemannian surfaces $Σ$ of bounded Euler genus has a triangulation $G\subset Σ$ such that $G^{(1)} \hookrightarrow Σ$ is a quasi-isometry, where $G^{(1)}$ is the simplicial 1-skeleton of $G$.

math.CO↗
arXiv2026

Surfaces without quasi-isometric simplicial triangulations

We construct a complete Riemannian surface $Σ$ that admits no triangulation $G\subset Σ$ such that the inclusion $G^{(1)} \hookrightarrow Σ$ is a quasi-isometry, where $G^{(1)}$ is the simplicial 1-skeleton of $G$. Our construction is without boundary, has arbitrarily large systole, and furthermore, there is no embedded graph $G\subsetΣ$ such that $G^{(1)} \hookrightarrow Σ$ is a quasi-isometry. This answers a question of Georgakopoulos.

math.MG↗
arXiv2026

Binary matroids and degree-boundedness for pivot-minors

We prove that for every bipartite graph $H$ and positive integer $s$, the class of $K_{s,s}$-subgraph-free graphs excluding $H$ as a pivot-minor has bounded average degree. Our proof relies on the announced binary matroid structure theorem of Geelen, Gerards, and Whittle. Along the way, we also prove that every $K_{s,t}$-free bipartite circle graph with $s\le t$ has a vertex of degree at most $\max\{2s-2, t-1\}$ and provide examples showing that this is tight.

math.CO↗
arXiv2026

Counterexample to the conjectured coarse grid theorem

We show that for every $M,A,n \in \mathbb{N}$ there exists a graph $G$ that does not contain the $(154\times 154)$-grid as a $3$-fat minor and is not $(M,A)$-quasi-isometric to a graph with no $K_n$ minor. This refutes the conjectured coarse grid theorem by Georgakopoulos and Papasoglu and the weak fat minor conjecture of Davies, Hickingbotham, Illingworth, and McCarty. Our construction is a slight modification of the recent counterexample to the weak coarse Menger conjecture from Nguyen, Scott and Seymour. We further modify the construction to show that there are planar graphs that do not have the coarse Erdős-Pósa property.

math.CO↗
arXiv2026

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 $χ_{\mathrm{o}}$-bounded.

math.CO↗
arXiv2026

Reuniting $χ$-boundedness with polynomial $χ$-boundedness

A class $\mathcal{F}$ of graphs is $χ$-bounded if there is a function $f$ such that $χ(H)\le f(ω(H))$ for all induced subgraphs $H$ of a graph in $\mathcal{F}$. If $f$ can be chosen to be a polynomial, we say that $\mathcal{F}$ is polynomially $χ$-bounded. Esperet proposed a conjecture that every $χ$-bounded class of graphs is polynomially $χ$-bounded. This conjecture has been disproved; it has been shown that there are classes of graphs that are $χ$-bounded but not polynomially $χ$-bounded. Nevertheless, inspired by Esperet's conjecture, we introduce Pollyanna classes of graphs. A class $\mathcal{C}$ of graphs is Pollyanna if $\mathcal{C}\cap \mathcal{F}$ is polynomially $χ$-bounded for every $χ$-bounded class $\mathcal{F}$ of graphs. We prove that several classes of graphs are Pollyanna and also present some proper classes of graphs that are not Pollyanna.

math.CO↗
arXiv2025

Preparing graph states forbidding a vertex-minor

Measurement based quantum computing is preformed by adding non-Clifford measurements to a prepared stabilizer states. Entangling gates like CZ are likely to have lower fidelities due to the nature of interacting qubits, so when preparing a stabilizer state, we wish to minimize the number of required entangling states. This naturally introduces the notion of CZ-distance. Every stabilizer state is local-Clifford equivalent to a graph state, so we may focus on graph states $\left\vert G \right\rangle$. As a lower bound for general graphs, there exist $n$-vertex graphs $G$ such that the CZ-distance of $\left\vert G \right\rangle$ is $Ω(n^2 / \log n)$. We obtain significantly improved bounds when $G$ is contained within certain proper classes of graphs. For instance, we prove that if $G$ is a $n$-vertex circle graph with clique number $ω$, then $\left\vert G \right\rangle$ has CZ-distance at most $4n \log ω+ 7n$. We prove that if $G$ is an $n$-vertex graph of rank-width at most $k$, then $\left\vert G \right\rangle$ has CZ-distance at most $(2^{2^{k+1}} + 1) n$. More generally, this is obtained via a bound of $(k+2)n$ that we prove for graphs of twin-width at most $k$. We also study how bounded-rank perturbations and low-rank cuts affect the CZ-distance. As a consequence, we prove that Geelen's Weak Structural Conjecture for vertex-minors implies that if $G$ is an $n$-vertex graph contained in some fixed proper vertex-minor-closed class of graphs, then $\left\vert G \right\rangle$ has CZ-distance at most $O(n\log n)$. Since graph states of locally equivalent graphs are local Clifford equivalent, proper vertex-minor-closed classes of graphs are natural and very general in this setting.

quant-ph↗
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.

math.CO↗
arXiv2025

Pivot-minors and the Erdős-Hajnal conjecture

We prove a conjecture of Kim and Oum that every proper pivot-minor-closed class of graphs has the strong Erdős-Hajnal property. More precisely, for every graph $H$, there exists $ε> 0$ such that every $n$-vertex graph with no pivot-minor isomorphic to $H$ contains two sets $A, B$ of vertices such that $|A|, |B| \ge εn$ and $A$ is complete or anticomplete to $B$.

math.CO↗