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9084 results in Discrete Mathematics, Information Theory and Coding

Page 44 of 455

  • On tripartite common graphs

  • Part of
  • Andrzej Grzesik, Joonkyung Lee, Bernard Lidický, Jan Volec
  • Journal:
    Combinatorics, Probability and Computing / Volume 31 / Issue 5 / September 2022
    Published online by Cambridge University Press:
    25 May 2022, pp. 907-923
    • Article
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  • A graph $H$ is common if the number of monochromatic copies of $H$ in a 2-edge-colouring of the complete graph $K_n$ is asymptotically minimised by the random colouring. Burr and Rosta, extending a famous conjecture of Erdős, conjectured that every graph is common. The conjectures of Erdős and of Burr and Rosta were disproved by Thomason and by Sidorenko, respectively, in the late 1980s. Collecting new examples of common graphs had not seen much progress since then, although very recently a few more graphs were verified to be common by the flag algebra method or the recent progress on Sidorenko’s conjecture. Our contribution here is to provide several new classes of tripartite common graphs. The first example is the class of so-called triangle trees, which generalises two theorems by Sidorenko and answers a question of Jagger, Šťovíček, and Thomason from 1996. We also prove that, somewhat surprisingly, given any tree $T$, there exists a triangle tree such that the graph obtained by adding $T$ as a pendant tree is still common. Furthermore, we show that adding arbitrarily many apex vertices to any connected bipartite graph on at most $5$ vertices yields a common graph.

Surveys in Combinatorics 2022
  • Edited by Anthony Nixon, Sean Prendiville
  • Published online:
    13 May 2022
    Print publication:
    09 June 2022
  • This volume contains eight survey articles by the invited speakers of the 29th British Combinatorial Conference, held at Lancaster University in July 2022. Each article provides an overview of recent developments in a current hot research topic in combinatorics. These topics span graphs and hypergraphs, Latin squares, linear programming, finite fields, extremal combinatorics, Ramsey theory, graph minors and tropical geometry. The authors are among the world's foremost researchers on their respective topics but their surveys are aimed at nonspecialist readers: they are written clearly with little prior knowledge assumed and with pointers to the wider literature. Taken together these surveys give a snapshot of the research frontier in contemporary combinatorics, making the latest developments accessible to researchers and graduate students in mathematics and theoretical computer science with an interest in combinatorics and helping them to keep abreast of the field.

Page 44 of 455