### Abstract

For a positive integer n and a graphical property P, ext_{n}(P) will denote the minimal number of edges in a graph G of order n that satisfies property P, and Ext>_{n}(P) will denote the class of graphs with n vertices and ext_{n}(P) edges that have property P. The extremal numbers ext_{n}(P) for properties P that have been commonly used as sufficient conditions for Hamiltonian paths and cycles in graphs will be investigated. In particular, results on the extremal numbers for the generalized degree and generalized independent degree properties will be given, where for a fixed positive integer t, the generalized degree δ_{t}(G) (generalized independent degree δ^{i}_{t}(G)) is the minimum number of vertices in the union of the neighborhoods of a set of t (independent) vertices of the graph G.

Original language | English |
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Pages (from-to) | 139-152 |

Number of pages | 14 |

Journal | Discrete Mathematics |

Volume | 127 |

Issue number | 1-3 |

DOIs | |

Publication status | Published - Mar 15 1994 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*127*(1-3), 139-152. https://doi.org/10.1016/0012-365X(92)00473-5

**Extremal problems and generalized degrees.** / Erdős, P.; Faudree, Ralph J.; Rousseau, Cecil C.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 127, no. 1-3, pp. 139-152. https://doi.org/10.1016/0012-365X(92)00473-5

}

TY - JOUR

T1 - Extremal problems and generalized degrees

AU - Erdős, P.

AU - Faudree, Ralph J.

AU - Rousseau, Cecil C.

PY - 1994/3/15

Y1 - 1994/3/15

N2 - For a positive integer n and a graphical property P, extn(P) will denote the minimal number of edges in a graph G of order n that satisfies property P, and Ext>n(P) will denote the class of graphs with n vertices and extn(P) edges that have property P. The extremal numbers extn(P) for properties P that have been commonly used as sufficient conditions for Hamiltonian paths and cycles in graphs will be investigated. In particular, results on the extremal numbers for the generalized degree and generalized independent degree properties will be given, where for a fixed positive integer t, the generalized degree δt(G) (generalized independent degree δit(G)) is the minimum number of vertices in the union of the neighborhoods of a set of t (independent) vertices of the graph G.

AB - For a positive integer n and a graphical property P, extn(P) will denote the minimal number of edges in a graph G of order n that satisfies property P, and Ext>n(P) will denote the class of graphs with n vertices and extn(P) edges that have property P. The extremal numbers extn(P) for properties P that have been commonly used as sufficient conditions for Hamiltonian paths and cycles in graphs will be investigated. In particular, results on the extremal numbers for the generalized degree and generalized independent degree properties will be given, where for a fixed positive integer t, the generalized degree δt(G) (generalized independent degree δit(G)) is the minimum number of vertices in the union of the neighborhoods of a set of t (independent) vertices of the graph G.

UR - http://www.scopus.com/inward/record.url?scp=10644294247&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=10644294247&partnerID=8YFLogxK

U2 - 10.1016/0012-365X(92)00473-5

DO - 10.1016/0012-365X(92)00473-5

M3 - Article

AN - SCOPUS:10644294247

VL - 127

SP - 139

EP - 152

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 1-3

ER -