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James Davies

Publications and source records attributed to James Davies.

At least 37 records · Page 2Linked to original sources
arXiv2024

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, Chvátal 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 $ω$ is $(ω+ 199050)$-colourable.

math.CO↗
arXiv2024

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, Hliněný 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 $Σ$, a connected graph $G$ has a finite cover embeddable in $Σ$ if and only if $G$ is embeddable in a surface covered by $Σ$. As evidence toward this, we prove that for every surface $Σ$, the connected graphs with a finite cover embeddable in $Σ$ 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 $Σ$, there exists a decreasing function $p_Σ:\mathbb{N} \to \mathbb{N}$ with $\lim_{g\to \infty}p_Σ(g) =0$ such that every finite cover embeddable in $Σ$ of any connected graph with Euler genus $g\ge c$ has ply at most $p_Σ(g)$.

math.CO↗
arXiv2024

Chromatic number of spacetime

We observe that an old theorem of Graham implies that for any positive integer $s$, there exists some positive integer $T(s)$ such that every $s$-colouring of $\mathbb{Z}^2$ contains a monochromatic pair of points $(x,y),(x',y')$ with $(x-x')^2 - (y-y')^2 = (T(s))^2$. By scaling, this implies that every finite colouring of $\mathbb{Q}^2$ contains a monochromatic pair of points $(x,y),(x',y')$ with $(x-x')^2 - (y-y')^2 = 1$, which answers in a strong sense a problem of Kosheleva and Kreinovich on a pseudo-Euclidean analogue of the Hadwiger-Nelson problem. The proof of Graham's theorem relies on repeated applications of van der Waerden's theorem, and so the resulting function $T(s)$ grows extremely quickly. We give an alternative proof in the weaker setting of having a second spacial dimension that results in a significantly improved bound. To be more precise, we prove that for every positive integer $s$ with $r\equiv 2 \pmod{4}$, every $s$-colouring of $\mathbb{Z}^3$ contains a monochromatic pair of points $(x,y,z),(x',y',z')$ such that $(x-x')^2 + (y-y')^2 - (z-z')^2 = (5^{(s-2)/4}(8\cdot 5^{(s-2)/2})!)^2$. In fact, we prove a stronger density version. The density version in $\mathbb{Z}^2$ remains open.

math.CO↗
arXiv2024

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.

math.CO↗
arXiv2024

Geometric graphs with exponential chromatic number and arbitrary girth

In 1975 Erdős initiated the study of the following very natural question. What can be said about the chromatic number of unit distance graphs in $\mathbb{R}^2$ that have large girth? Over the years this question and its natural extension to $\mathbb{R}^d$ attracted considerable attention with the high-dimensional variant reiterated recently by Alon and Kupavskii. We prove that there exist unit distance graphs in $\mathbb{R}^d$ with chromatic number at least $(1.074 + o(1))^d$ that have arbitrarily large girth. This improves upon a series of results due to Kupavskii; Sagdeev; and Sagdeev and Raigorodskii and gives the first bound in which the base of the exponent does not tend to one with the girth. In addition, our construction can be made explicit which allows us to answer in a strong form a question of Kupavskii. Our arguments show graphs of large chromatic number and high girth exist in a number of other geometric settings including diameter graphs and orthogonality graphs.

math.CO↗
arXiv2024

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.

math.CO↗
arXiv2024

On polynomial degree-boundedness

We prove a conjecture of Bonamy, Bousquet, Pilipczuk, Rzążewski, Thomassé, and Walczak, that for every graph $H$, there is a polynomial $p$ such that for every positive integer $s$, every graph of average degree at least $p(s)$ contains either $K_{s,s}$ as a subgraph or contains an induced subdivision of $H$. This improves upon a result of Kühn and Osthus from 2004 who proved it for graphs whose average degree is at least triply exponential in $s$ and a recent result of Du, Girão, Hunter, McCarty and Scott for graphs with average degree at least singly exponential in $s$. As an application, we prove that the class of graphs that do not contain an induced subdivision of $K_{s,t}$ is polynomially $χ$-bounded. In the case of $K_{2,3}$, this is the class of theta-free graphs, and answers a question of Davies. Along the way, we also answer a recent question of McCarty, by showing that if $\mathcal{G}$ is a hereditary class of graphs for which there is a polynomial $p$ such that every bipartite $K_{s,s}$-free graph in $\mathcal{G}$ has average degree at most $p(s)$, then more generally, there is a polynomial $p'$ such that every $K_{s,s}$-free graph in $\mathcal{G}$ has average degree at most $p'(s)$. Our main new tool is an induced variant of the Kővári-Sós-Turán theorem, which we find to be of independent interest.

math.CO↗
arXiv2024

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.

math.CO↗
arXiv2024

Fat minors cannot be thinned (by quasi-isometries)

We disprove the conjecture of Georgakopoulos and Papasoglu that a length space (or graph) with no $K$-fat $H$ minor is quasi-isometric to a graph with no $H$ minor. Our counterexample is furthermore not quasi-isometric to a graph with no 2-fat $H$ minor or a length space with no $H$ minor. On the other hand, we show that the following weakening holds: any graph with no $K$-fat $H$ minor is quasi-isometric to a graph with no $3$-fat $H$ minor.

math.CO↗
arXiv2024

Smart Textile-Driven Soft Spine Exosuit for Lifting Tasks in Industrial Applications

Work related musculoskeletal disorders (WMSDs) are often caused by repetitive lifting, making them a significant concern in occupational health. Although wearable assist devices have become the norm for mitigating the risk of back pain, most spinal assist devices still possess a partially rigid structure that impacts the user comfort and flexibility. This paper addresses this issue by presenting a smart textile actuated spine assistance robotic exosuit (SARE), which can conform to the back seamlessly without impeding the user movement and is incredibly lightweight. The SARE can assist the human erector spinae to complete any action with virtually infinite degrees of freedom. To detect the strain on the spine and to control the smart textile automatically, a soft knitting sensor which utilizes fluid pressure as sensing element is used. The new device is validated experimentally with human subjects where it reduces peak electromyography (EMG) signals of lumbar erector spinae by around 32 percent in loaded and around 22 percent in unloaded conditions. Moreover, the integrated EMG decreased by around 24.2 percent under loaded condition and around 23.6 percent under unloaded condition. In summary, the artificial muscle wearable device represents an anatomical solution to reduce the risk of muscle strain, metabolic energy cost and back pain associated with repetitive lifting tasks.

cs.RO↗
arXiv2023

A solution to Ringel's circle problem

We construct families of circles in the plane such that their tangency graphs have arbitrarily large girth and chromatic number. This provides a strong negative answer to Ringel's circle problem (1959). The proof relies on a (multidimensional) version of Gallai's theorem with polynomial constraints, which we derive from the Hales-Jewett theorem and which may be of independent interest.

math.CO↗
arXiv2023

Constraints on axionlike particles from a combined analysis of three flaring $\textit{Fermi}$ flat-spectrum radio quasars

Many theories beyond the Standard Model of particle physics predict the existence of axionlike particles (ALPs) that mix with photons in the presence of a magnetic field. Searching for the effects of ALP-photon mixing in gamma-ray observations of blazars has provided some of the strongest constraints on ALP parameter space so far. Previously, only individual sources have been analysed. We perform a combined analysis on $\textit{Fermi}$ Large Area Telescope data of three bright, flaring flat-spectrum radio quasars, with the blazar jets themselves as the dominant mixing region. For the first time, we include a full treatment of photon-photon dispersion within the jet, and account for the uncertainty in our B-field model by leaving the field strength free in the fitting. Overall, we find no evidence for ALPs, but are able to exclude the ALP parameters $m_a\lesssim200$ neV and $g_{aγ}\gtrsim 5 \times 10^{-12}$ GeV$^{-1}$ with 95\% confidence.

astro-ph.HE↗
arXiv2022

Separating polynomial $χ$-boundedness from $χ$-boundedness

Extending the idea from the recent paper by Carbonero, Hompe, Moore, and Spirkl, for every function $f\colon\mathbb{N}\to\mathbb{N}\cup\{\infty\}$ with $f(1)=1$ and $f(n)\geq\binom{3n+1}{3}$, we construct a hereditary class of graphs $\mathcal{G}$ such that the maximum chromatic number of a graph in $\mathcal{G}$ with clique number $n$ is equal to $f(n)$ for every $n\in\mathbb{N}$. In particular, we prove that there exist hereditary classes of graphs that are $χ$-bounded but not polynomially $χ$-bounded.

math.CO↗
arXiv2022

Triangle-free graphs with large chromatic number and no induced wheel

A wheel is a graph consisting of an induced cycle of length at least four and a single additional vertex with at least three neighbours on the cycle. We prove that no Burling graph contains an induced wheel. Burling graphs are triangle-free and have arbitrarily large chromatic number, so this answers a question of Trotignon and disproves a conjecture of Scott and Seymour.

math.CO↗
arXiv2022

Improved bounds for colouring circle graphs

We prove the first $χ$-bounding function for circle graphs that is optimal up to a constant factor. To be more precise, we prove that every circle graph with clique number at most $ω$ has chromatic number at most $2ω\log_2 (ω) +2ω\log_2(\log_2 (ω)) + 10ω$.

math.CO↗
arXiv2021

Relevance of photon-photon dispersion within the jet for blazar axionlike particle searches

Axionlike particles (ALPs) could mix with photons in the presence of astrophysical magnetic fields. Searching for this effect in gamma-ray observations of blazars has provided some of the strongest constraints on ALP parameter space so far. Previously, photon-photon dispersion of gamma-rays off of the CMB has been shown to be important for these calculations, and is universally included in ALP-photon mixing models. Here, we assess the effects of dispersion off of other photon fields within the blazar (produced by the accretion disk, the broad line region, the dust torus, starlight, and the synchrotron field) by modelling the jet and fields of the flat spectrum radio quasar 3C454.3 and propagating ALPs through the model both with and without the full dispersion calculation. We find that the full dispersion calculation can strongly affect the mixing, particularly at energies above 100 GeV -- often reducing the ALP-photon conversion probability. This could have implications for future searches planned with, e.g., the Cherenkov Telescope Array, particularly those looking for a reduced opacity of the universe at the highest energies.

astro-ph.HE↗