The Maximum Number of Complete Subgraphs of Fixed Size in a Graph with Given Maximum Degree

Jonathan Cutler, A. J. Radcliffe

Research output: Contribution to journalArticle

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In this article, we make progress on a question related to one of Galvin that has attracted substantial attention recently. The question is that of determining among all graphs G with n vertices and Δ(G) ≤ r, which has the most complete subgraphs of size t, for t≥3. The conjectured extremal graph is aKr+1 ∪ Kb, where n = a(r + 1) + b with 0 ≤ b ≤ r. Gan et al. (Combin Probab Comput 24(3) (2015), 521–527) proved the conjecture when a ≤ 1, and also reduced the general conjecture to the case t = 3. We prove the conjecture for r ≤ 6 and also establish a weaker form of the conjecture for all r.

Original languageEnglish
Pages (from-to)134-145
Number of pages12
JournalJournal of Graph Theory
Issue number2
StatePublished - 1 Feb 2017



  • cliques
  • complete subgraphs
  • independent sets

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