### Abstract

We consider directed graphs without loops and multiple edges, where the exclusion of multiple edges means that two vertices cannot be joined by two edges of the same orientation. Let L_{1},...,L_{a} be given digraphs. What is the maximum number of edges a digraph can have if it does not contain and L_{i} as a subgraph and has given number of vertices? We shall prove the existence of a sequence of asymptotical extremal graphs having fairly simple structure. More exactly:. There exist a matrix A = (a_{i.j})_{i.j≤r} and a sequence {S^{n}} of graphs such that 1. (i) the vertices of S^{n} can be divided into classes C_{1},...,C_{r} so that, if i ≠ j, each vertex of C_{i} is joined to each vertex of C_{j} by an edge oriented from C_{i} to C_{j} if and only if a_{i.j} = 2; the vertices of C_{i} are independent if a_{i.i} = 0; and otherwise a_{i.i} = 1 and the digraph determined by C_{i} is a complete acyclic digraph; 2. (ii) S^{n} contains no L_{i} but any graph having [∈n^{2}] more edges than S^{n} must contain at least one L_{i}. (Here the word graph is an "abbreviation" for "directed graph or digraph.").

Original language | English |
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Pages (from-to) | 77-93 |

Number of pages | 17 |

Journal | Journal of Combinatorial Theory, Series B |

Volume | 15 |

Issue number | 1 |

DOIs | |

Publication status | Published - Aug 1973 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

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## Cite this

*Journal of Combinatorial Theory, Series B*,

*15*(1), 77-93. https://doi.org/10.1016/0095-8956(73)90034-8