Connected Colourings of Complete Graphs and Hypergraphs

Abstract

Gallai’s colouring theorem states that if the edges of a complete graph are 3-coloured, with each colour class forming a connected (spanning) subgraph, then there is a triangle that has all three colours. What happens for more colours: if we \(k\)-colour the edges of the complete graph, with each colour class connected, how many of the \(\left( {\begin{array}{c}k\\ 3\end{array}}\right) \) triples of colours must appear as triangles? In this note we show that the ‘obvious’ conjecture, namely that there are always at least \(\left( {\begin{array}{c}k-1\\ 2\end{array}}\right) \) triples, is not correct. We determine the minimum asymptotically. This answers a question of Johnson. We also give some results about the analogous problem for hypergraphs, and we make a conjecture that we believe is the ‘right’ generalisation of Gallai’s theorem to hypergraphs.

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Correspondence to Ta Sheng Tan.

Additional information

Ta Sheng Tan was supported by the University Malaya Research Fund Assistance (BKP) via grant BK021-2013.

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Leader, I., Tan, T.S. Connected Colourings of Complete Graphs and Hypergraphs. Graphs and Combinatorics 32, 257–269 (2016). https://doi.org/10.1007/s00373-015-1537-2

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Keywords

  • Gallai colourings
  • Graph colourings
  • Hypergraphs
  • Extremal graph theory

Mathematics Subject Classification

  • 05C15
  • 05C65