Kirsch, Rachel and Radcliffe, A. J.
(2019)
*Many triangles with few edges.*
Electronic Journal of Combinatorics, 26 (2).
ISSN 1077-8926

Text (Many triangles with few edges)
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## Abstract

Extremal problems concerning the number of independent sets or complete subgraphs in a graph have been well studied in recent years. Cutler and Radcliffe proved that among graphs with n vertices and maximum degree at most r, where n = a(r + 1) + b and 0 ≤ b ≤ r, aKr+1 ∪ Kb has the maximum number of complete subgraphs, answering a question of Galvin. Gan, Loh and Sudakov conjectured that aKr+1 ∪Kb also maximizes the number of complete subgraphs Kt for each fixed size t ≥3, and proved this for a = 1. Cutler and Radcliffe proved this conjecture for r ≤ 6. We investigate a variant of this problem where we fix the number of edges instead of the number of vertices. We prove that aKr+1 ∪C(b), where C(b) is the colex graph on b edges, maximizes the number of triangles among graphs with m edges and any fixed maximum degree r ≤ 8, where m = a(r+1 2 ) + b and 0 ≤ b < (r+1 2 ). Mathematics Subject Classifications: 05.

Item Type: | Article |
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Official URL: | https://www.combinatorics.org/ |

Additional Information: | © 2019 The Authors |

Divisions: | Mathematics |

Subjects: | Q Science > QA Mathematics |

Date Deposited: | 10 Jul 2019 12:45 |

Last Modified: | 20 May 2021 00:25 |

URI: | http://eprints.lse.ac.uk/id/eprint/101147 |

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