Excluding a small minor

There are sixteen 3-connected graphs on eleven or fewer edges. For each of these graphs H we discuss the structure of graphs that do not contain a minor isomorphic to H. © 2012 Elsevier B.V. All rights reserved

Defective and Clustered Graph Colouring

Consider the following two ways to colour the vertices of a graph where the requirement that adjacent vertices get distinct colours is relaxed. A colouring has "defect" $d$ if each monochromatic component has maximum degree at most $d$. A colouring has "clustering" $c$ if each monochromatic component has at most $c$ vertices. This paper surveys research on these types of colourings, where the first priority is to minimise the number of colours, with small defect or small clustering as a secondary goal. List colouring variants are also considered. The following graph classes are studied: outerplanar graphs, planar graphs, graphs embeddable in surfaces, graphs with given maximum degree, graphs with given maximum average degree, graphs excluding a given subgraph, graphs with linear crossing number, linklessly or knotlessly embeddable graphs, graphs with given Colin de Verdi\`ere parameter, graphs with given circumference, graphs excluding a fixed graph as an immersion, graphs with given thickness, graphs with given stack- or queue-number, graphs excluding $K_t$ as a minor, graphs excluding $K_{s,t}$ as a minor, and graphs excluding an arbitrary graph $H$ as a minor. Several open problems are discussed.Comment: This is a preliminary version of a dynamic survey to be published in the Electronic Journal of Combinatoric

Low Energy Supersymmetry from Non-Geometry

We study a class of flux compactifications that have all the moduli stabilised, a high (GUT) string scale and a low (TeV) gravitino mass that is generated dynamically. These non-geometric compactifications correspond to type II string theories on SU(3)xSU(3) structure orientifolds. The resulting superpotentials admit, excluding non-perturbative effects, supersymmetric Minkowski vacua with any number of moduli stabilised. We argue that non-perturbative effects are present and introduce terms in the superpotential that are exponentially suppressed by the same moduli that appear perturbatively. These deform the supersymmetric Minkowski vacua to supersymmetric AdS vacua with an exponentially small gravitino mass. The resulting vacua allow for low scale supersymmetry breaking which can be realised by a number of mechanisms.Comment: 36pp; v2 references added, minor clarifications, JHEP versio

Primordial Black Holes, Eternal Inflation, and the Inflationary Parameter Space after WMAP5

We consider constraints on inflation driven by a single, minimally coupled scalar field in the light of the WMAP5 dataset, as well as ACBAR and the SuperNova Legacy Survey. We use the Slow Roll Reconstruction algorithm to derive optimal constraints on the inflationary parameter space. The scale dependence in the slope of the scalar spectrum permitted by WMAP5 is large enough to lead to viable models where the small scale perturbations have a substantial amplitude when extrapolated to the end of inflation. We find that excluding parameter values which would cause the overproduction of primordial black holes or even the onset of eternal inflation leads to potentially significant constraints on the slow roll parameters. Finally, we present a more sophisticated approach to including priors based on the total duration of inflation, and discuss the resulting restrictions on the inflationary parameter space.Comment: v2: version published in JCAP. Minor clarifications and references adde

Excluding Electroweak Baryogenesis in the MSSM

In the context of the MSSM the Light Stop Scenario (LSS) is the only region of parameter space that allows for successful Electroweak Baryogenesis (EWBG). This possibility is very phenomenologically attractive, since it allows for the direct production of light stops and could be tested at the LHC. The ATLAS and CMS experiments have recently supplied tantalizing hints for a Higgs boson with a mass of ~ 125 GeV. This Higgs mass severely restricts the parameter space of the LSS, and we discuss the specific predictions made for EWBG in the MSSM. Combining data from all the available ATLAS and CMS Higgs searches reveals a tension with the predictions of EWBG even at this early stage. This allows us to exclude EWBG in the MSSM at greater than (90) 95% confidence level in the (non-)decoupling limit, by examining correlations between different Higgs decay channels. We also examine the exclusion without the assumption of a ~ 125 GeV Higgs. The Higgs searches are still highly constraining, excluding the entire EWBG parameter space at greater than 90% CL except for a small window of m_h ~ 117 - 119 GeV.Comment: 24 Pages, 4 Figures (v3: fixed typos, minor corrections, added references

The Generalised Colouring Numbers on Classes of Bounded Expansion

The generalised colouring numbers $\mathrm{adm}_r(G)$, $\mathrm{col}_r(G)$, and $\mathrm{wcol}_r(G)$ were introduced by Kierstead and Yang as generalisations of the usual colouring number, also known as the degeneracy of a graph, and have since then found important applications in the theory of bounded expansion and nowhere dense classes of graphs, introduced by Ne\v{s}et\v{r}il and Ossona de Mendez. In this paper, we study the relation of the colouring numbers with two other measures that characterise nowhere dense classes of graphs, namely with uniform quasi-wideness, studied first by Dawar et al. in the context of preservation theorems for first-order logic, and with the splitter game, introduced by Grohe et al. We show that every graph excluding a fixed topological minor admits a universal order, that is, one order witnessing that the colouring numbers are small for every value of $r$. Finally, we use our construction of such orders to give a new proof of a result of Eickmeyer and Kawarabayashi, showing that the model-checking problem for successor-invariant first-order formulas is fixed-parameter tractable on classes of graphs with excluded topological minors

Upper limits on neutrino masses from the 2dFGRS and WMAP: the role of priors

Solar, atmospheric, and reactor neutrino experiments have confirmed neutrino oscillations, implying that neutrinos have non-zero mass, but without pinning down their absolute masses. While it is established that the effect of neutrinos on the evolution of cosmic structure is small, the upper limits derived from large-scale structure data could help significantly to constrain the absolute scale of the neutrino masses. In a recent paper the 2dF Galaxy Redshift Survey (2dFGRS) team provided an upper limit m_nu,tot < 2.2 eV, i.e. approximately 0.7 eV for each of the three neutrino flavours, or phrased in terms of their contributioin to the matter density, Omega_nu/Omega_m < 0.16. Here we discuss this analysis in greater detail, considering issues of assumed 'priors' like the matter density Omega_m and the bias of the galaxy distribution with respect the dark matter distribution. As the suppression of the power spectrum depends on the ratio Omega_nu/Omega_m, we find that the out-of- fashion Mixed Dark Matter Model, with Omega_nu=0.2, Omega_m=1 and no cosmological constant, fits the 2dFGRS power spectrum and the CMB data reasonably well, but only for a Hubble constant H_0<50 km/s/Mpc. As a consequence, excluding low values of the Hubble constant, e.g. with the HST Key Project is important in order to get a strong constraint on the neutrino masses. We also comment on the improved limit by the WMAP team, and point out that the main neutrino signature comes from the 2dFGRS and the Lyman alpha forest.Comment: 24 pages, 12 figures Minor changes to matched version published in JCA