### Abstract

This chapter presents a study on graphs which contain all sparse graphs. Let ℋ_{n} denote the class of all graphs with n edges and denote by s(n) the minimum number of edges a graph G can have, which contains all H ∊ ℋ_{n} as subgraphs. The chapter discusses the problem of determining the minimum number of edges, denoted by s'(n) a graph can have, which contains every planar graph on n edges as a subgraph. The chapter discusses a lower bound for s(n) and while discussing an upper bound for s(n), it is proved (by the probability method) that there exists a graph with cn^{2} log log n/log n edges that contains all graphs with at most n edges. The chapter presents a theorem to give an upper bound of n^{3/2} for the universal graphs that contain all planar graphs on n edges.

Original language | English (US) |
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Pages (from-to) | 21-26 |

Number of pages | 6 |

Journal | North-Holland Mathematics Studies |

Volume | 60 |

Issue number | C |

DOIs | |

State | Published - Jan 1 1982 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*North-Holland Mathematics Studies*,

*60*(C), 21-26. https://doi.org/10.1016/S0304-0208(08)73486-8