### Abstract

Let ð�’¢(n, m) denote the class of simple graphs on n vertices and m edges and let G ∈ ð�’¢ (n, m). There are many results in graph theory giving conditions under which G contains certain types of subgraphs, such as cycles of given lengths, complete graphs, etc. For example, Turan's theorem gives a sufficient condition for G to contain a K_{k + 1} in terms of the number of edges in G. In this paper we prove that, for m = αn^{2}, α > (k ‐ 1)/2k, G contains a K_{k + 1}, each vertex of which has degree at least f(α)n and determine the best possible f(α). For m = ⌊n^{2}/4⌋ + 1 we establish that G contains cycles whose vertices have certain minimum degrees. Further, for m = αn^{2}, α > 0 we establish that G contains a subgraph H with δ(H) ≥ f(α, n) and determine the best possible value of f(α, n).

Original language | English |
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Pages (from-to) | 17-27 |

Number of pages | 11 |

Journal | Journal of Graph Theory |

Volume | 12 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1988 |

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### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Journal of Graph Theory*,

*12*(1), 17-27. https://doi.org/10.1002/jgt.3190120104