Minimal Cayley graphs with large chromatic number
Resolving Babai's minimal Cayley graph problem, we construct finite minimal Cayley graphs with arbitrarily large chromatic number.
math.CO↗
arXiv subjects
James Davies
Publications and source records attributed to James Davies.
On the chromatic number of the union of comparability graphs
Resolving in a strong sense a problem of Gy\'arf\'as 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 $\frac{|V(G)|}{\alpha(G)}$.
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Surfaces without quasi-isometric simplicial triangulations
We construct a complete Riemannian surface $\Sigma$ that admits no triangulation $G\subset \Sigma$ such that the inclusion $G^{(1)} \hookrightarrow \Sigma$ 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 $G\subset\Sigma$ such that $G^{(1)} \hookrightarrow \Sigma$ is a quasi-isometry. This answers a question of Georgakopoulos.
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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 $\Gamma$ is virtually a free product of free and surface groups if and only if $\Gamma$ 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 $\Sigma$ of bounded Euler genus has a triangulation $G\subset \Sigma$ such that $G^{(1)} \hookrightarrow \Sigma$ is a quasi-isometry, where $G^{(1)}$ is the simplicial 1-skeleton of $G$.
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Burling graphs in graphs with large chromatic number
A graph class is $\chi$-bounded if the only way to force large chromatic number in graphs from the class is by forming a large clique. In the 1970s, Erd\H{o}s conjectured that intersection graphs of straight-line segments in the plane are $\chi$-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 $\chi(B_k)=k$ 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 $\chi$-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 $\chi$-bounded.
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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 Erd\H{o}s-P\'{o}sa property.
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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.
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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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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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Strongly sublinear separators and bounded asymptotic dimension for sphere intersection graphs
In this paper, we consider the class $\mathcal{C}^d$ of sphere intersection graphs in $\mathbb{R}^d$ for $d \geq 2$. We show that for each integer $t$, the class of all graphs in $\mathcal{C}^d$ that exclude $K_{t,t}$ as a subgraph has strongly sublinear separators. We also prove that $\mathcal{C}^d$ has asymptotic dimension at most $2d+2$.
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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 $\Omega(n^2 / \log n)$. 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 $\omega$, then $\left\vert G \right\rangle$ has CZ-distance at most $4n \log \omega + 7n$. 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.
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Quasi-isometries between graphs with variable edge lengths
This paper investigates quasi-isometries between graphs with variable edge lengths. A quasi-isometry is a mapping between metric spaces that approximately preserves distances, allowing for a bounded amount of additive and multiplicative distortion. Recently, Nguyen, Scott, and Seymour conjectured that, by appropriately adjusting the edge lengths of the target graph along with modifying the additive distortion constant, the multiplicative distortion factor could be eliminated. We disprove this conjecture.
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Colouring t-perfect graphs
Perfect graphs can be described as the graphs whose stable set polytopes are defined by their non-negativity and clique inequalities (including edge inequalities). In 1975, Chv\'{a}tal defined an analogous class of t-perfect graphs, which are the graphs whose stable set polytopes are defined by their non-negativity, edge inequalities, and odd circuit inequalities. We show that t-perfect graphs are $199053$-colourable. This is the first finite bound on the chromatic number of t-perfect graphs and answers a question of Shepherd from 1995. Our proof also shows that every h-perfect graph with clique number $\omega$ is $(\omega + 199050)$-colourable.
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On high genus extensions of Negami's conjecture
Negami's famous planar cover conjecture is equivalent to the statement that a connected graph can be embedded in the projective plane if and only if it has a projective planar cover. In 1999, Hlin\v{e}n\'y proposed extending this conjecture to higher genus non-orientable surfaces. In this paper, we put forward a natural extension that encompasses orientable surfaces as well; for every compact surface $\Sigma$, a connected graph $G$ has a finite cover embeddable in $\Sigma$ if and only if $G$ is embeddable in a surface covered by $\Sigma$. As evidence toward this, we prove that for every surface $\Sigma$, the connected graphs with a finite cover embeddable in $\Sigma$ have bounded Euler genus. Moreover, we show that these extensions of Negami's conjecture are decidable for every compact surface of sufficiently large Euler genus, surpassing what is known for Negami's original conjecture. We also prove the natural analogue for countable graphs embeddable into a compact (orientable) surface. More precisely, we prove that a connected countable graph $G$ has a finite ply cover that embeds into a compact (orientable) surface if and only if $G$ embeds into a compact (orientable) surface. Our most general theorem, from which these results are derived, is that there is a constant $c>0$ such that for every surface $\Sigma$, there exists a decreasing function $p_\Sigma:\mathbb{N} \to \mathbb{N}$ with $\lim_{g\to \infty}p_\Sigma(g) =0$ such that every finite cover embeddable in $\Sigma$ of any connected graph with Euler genus $g\ge c$ has ply at most $p_\Sigma(g)$.
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Treewidth, Hadwiger Number, and Induced Minors
Treewidth and Hadwiger number are two of the most important parameters in structural graph theory. This paper studies graph classes in which large treewidth implies the existence of a large complete graph minor. To formalise this, we say that a graph class $\mathcal{G}$ is (tw,had)-bounded if there is a function $f$ (called the (tw,had)-bounding function) such that tw$(G)$ $\leq$ $f$(had$(G)$) for every graph $G \in \mathcal{G}$. We characterise (tw,had)-bounded graph classes as those that exclude some planar graph as an induced minor, and use this characterisation to show that every proper vertex-minor-closed class is (tw,had)-bounded. Furthermore, we demonstrate that any (tw,had)-bounded graph class has a (tw,had)-bounding function in O(had$(G)^9$polylog(had$(G)$)). Our bound comes from the bound for the Grid Minor Theorem given by Chuzhoy and Tan, and any quantitative improvement to their result will lead directly to an improvement to our result. More strongly, we conjecture that every (tw,had)-bounded graph class has a linear (tw,had)-bounding function. In support of this conjecture, we show that it holds for the class of outer-string graphs, and for a natural generalisation of outer-string graphs: intersection graphs of strings rooted at the boundary of a fixed surface. We also verify our conjecture for low-rank perturbations of circle graphs, which is an important step towards verifying it for all proper vertex-minor-closed classes.
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Counterexample to Babai's lonely colour conjecture
Motivated by colouring minimal Cayley graphs, in 1978, Babai conjectured that no-lonely-colour graphs have bounded chromatic number. We disprove this in a strong sense by constructing graphs of arbitrarily large girth and chromatic number that have a proper edge-colouring in which each cycle contains no colour exactly once.
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Polynomial Gyárfás-Sumner conjecture for graphs of bounded boxicity
We prove that for every positive integer $d$ and forest $F$, the class of intersection graphs of axis-aligned boxes in $\mathbb{R}^d$ with no induced $F$ subgraph is (polynomially) $χ$-bounded.
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Prime and polynomial distances in colourings of the plane
We give two extensions of the recent theorem of the first author that the odd distance graph has unbounded chromatic number. The first is that for any non-constant polynomial $f$ with integer coefficients and positive leading coefficient, every finite colouring of the plane contains a monochromatic pair of distinct points whose distance is equal to $f(n)$ for some integer $n$. The second is that for every finite colouring of the plane, there is a monochromatic pair of points whose distance is a prime number.
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