Extensions of a theorem of Erdős on nonhamiltonian graphs

Zoltán Füredi, Alexandr Kostochka, Ruth Luo

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

Let n,d be integers with 1 ≤ d ≤ [n-1/2], and set h(n,d):= (n-d/2)+ d2. Erdős proved that when n ≥ 6d, each n-vertex nonhamiltonian graph G with minimum degree δ(G) ≥ d has at most h(n,d) edges. He also provides a sharpness example Hn,d for all such pairs n,d. Previously, we showed a stability version of this result: for n large enough, every nonhamiltonian graph G on n vertices with δ(G) ≥ d and more than h(n,d+1) edges is a subgraph of Hn,d . In this article, we show that not only does the graph Hn,d maximize the number of edges among nonhamiltonian graphs with n vertices and minimum degree at least d, but in fact it maximizes the number of copies of any fixed graph F when n is sufficiently large in comparison with d and (Formula presented.). We also show a stronger stability theorem, that is, we classify all nonhamiltonian n-vertex graphs with δ(G) ≥ d and more than h(n,d +2) edges. We show this by proving a more general theorem: we describe all such graphs with more than (n-(d+2)/k)+ (d+2) (d+2 k-1) copies of Kk for any k.

Original languageEnglish
Pages (from-to)176-193
Number of pages18
JournalJournal of Graph Theory
Volume89
Issue number2
DOIs
Publication statusPublished - Oct 2018

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Keywords

  • extremal graph theory
  • hamiltonian cycles
  • subgraph density

ASJC Scopus subject areas

  • Geometry and Topology

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