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52669 results in Statistics and Probability

Page 263 of 2634

  • Clustered 3-colouring graphs of bounded degree

  • Part of
  • Vida Dujmović, Louis Esperet, Pat Morin, Bartosz Walczak, David R. Wood
  • Journal:
    Combinatorics, Probability and Computing / Volume 31 / Issue 1 / January 2022
    Published online by Cambridge University Press:
    18 June 2021, pp. 123-135

  • A (not necessarily proper) vertex colouring of a graph has clustering c if every monochromatic component has at most c vertices. We prove that planar graphs with maximum degree $\Delta$ are 3-colourable with clustering $O(\Delta^2)$. The previous best bound was $O(\Delta^{37})$. This result for planar graphs generalises to graphs that can be drawn on a surface of bounded Euler genus with a bounded number of crossings per edge. We then prove that graphs with maximum degree $\Delta$ that exclude a fixed minor are 3-colourable with clustering $O(\Delta^5)$. The best previous bound for this result was exponential in $\Delta$.

Statistics for Laboratory Scientists and Clinicians
  • A Practical Guide
  • Anne McDonnell Sill
  • Published online:
    17 June 2021
    Print publication:
    08 July 2021
  • Understanding the underlying principles of statistical techniques and effectively applying statistical methods can be challenging for researchers at all stages of their career. This concise, practical guide uses a simple, engaging approach to take scientists and clinicians working in laboratory-based life science and medical research through the steps of choosing and implementing appropriate statistical methods to analyse results. The author draws on her extensive experience of advising students and researchers over the past 30 years, breaking down complex concepts into easy-to-understand units. Practical examples using free online statistical tools are included throughout, with illustrations and diagrams employed to keep jargon to a minimum. Sample size calculations and considerations are covered in depth, and the book refers to types of data from experiments that clinicians and lab-based scientists are likely to encounter. Straightforward, accessible and encouraging throughout, this is a go-to reference for researchers who want to achieve statistical autonomy.

Page 263 of 2634