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2 results

  • 1 - Basic Graph Theory

  • Béla Bollobás, University of Cambridge, Robert Morris, IMPA, Rio de Janeiro
  • Book:
    Basic Graph Theory
    Published online:
    15 March 2026
    Print publication:
    26 March 2026, pp 1-43

  • Summary

    In this first chapter, we provide a gentle introduction to several basic but fundamental concepts in graph theory, such as paths, trees, connected components, independent sets and the chromatic number, illustrating each definition with some simple but important properties that will be used in later chapters. We also take the opportunity to introduce (in a very simple setting) the probabilistic method, which plays a central role in the book. In the second half of the chapter we introduce the reader to planar graphs, and provide a brief taste of some simple results from the areas of extremal graph theory and Ramsey theory, which are the central topics of Chapters 2 and 4, respectively.

  • LARGE $\mathcal {F}$-FREE SUBGRAPHS IN $r$-CHROMATIC GRAPHS

  • Part of
  • ZHEN HE, ZEQUN LV, XIUTAO ZHU
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 108 / Issue 2 / October 2023
    Published online by Cambridge University Press:
    02 December 2022, pp. 200-204
    Print publication:
    October 2023
    • Article
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  • For a graph G and a family of graphs $\mathcal {F}$, the Turán number ${\mathrm {ex}}(G,\mathcal {F})$ is the maximum number of edges an $\mathcal {F}$-free subgraph of G can have. We prove that ${\mathrm {ex}}(G,\mathcal {F})\ge {\mathrm {ex}}(K_r, \mathcal {F})$ if the chromatic number of G is r and $\mathcal {F}$ is a family of connected graphs. This result answers a question raised by Briggs and Cox [‘Inverting the Turán problem’, Discrete Math. 342(7) (2019), 1865–1884] about the inverse Turán number for all connected graphs.