### Abstract

The theory of extremal graphs without a fixed set of forbidden subgraphs is well developed. However, rather little is known about extremal graphs without forbidden subgraphs whose orders tend to ∞ with the order of the graph. In this note we deal with three problems of this latter type. Let L be a fixed bipartite graph and let L + E^{m} be the join of L with the empty graph of order m. As our first problem we investigate the maximum of the size e(G^{n}) of a graph G^{n} (i.e. a graph of order n) provided G^{n}⊅L + E[^{cn}, where c > 0 is a constant. In our second problem we study the maximum of e(G^{n}) if G^{ n}⊅K_{2}(r,cn) and G^{n}⊅ K^{3} . The third problem is of a slightly different nature. Let C^{k}(t) be obtained from a cycle C^{k} by multiplying each vertex by t. We shall prove that if c > 0 then there exists a constant l(c) such that if G^{n}⊅C^{k}(t) for k = 3, 5, 2l(c) + 1, then one can omit [cn^{2}] edges from G^{ n}so that the obtained graph is bipartite, provided n > n_{0} (c, t).

Original language | English |
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Pages (from-to) | 29-41 |

Number of pages | 13 |

Journal | Annals of Discrete Mathematics |

Volume | 3 |

Issue number | C |

DOIs | |

Publication status | Published - 1978 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

### Cite this

**Extremal Graphs without Large Forbidden Subgraphs.** / Bollobás, B.; Erdős, P.; Simonovits, M.; Szemerédi, E.

Research output: Contribution to journal › Article

*Annals of Discrete Mathematics*, vol. 3, no. C, pp. 29-41. https://doi.org/10.1016/S0167-5060(08)70495-3

}

TY - JOUR

T1 - Extremal Graphs without Large Forbidden Subgraphs

AU - Bollobás, B.

AU - Erdős, P.

AU - Simonovits, M.

AU - Szemerédi, E.

PY - 1978

Y1 - 1978

N2 - The theory of extremal graphs without a fixed set of forbidden subgraphs is well developed. However, rather little is known about extremal graphs without forbidden subgraphs whose orders tend to ∞ with the order of the graph. In this note we deal with three problems of this latter type. Let L be a fixed bipartite graph and let L + Em be the join of L with the empty graph of order m. As our first problem we investigate the maximum of the size e(Gn) of a graph Gn (i.e. a graph of order n) provided Gn⊅L + E[cn, where c > 0 is a constant. In our second problem we study the maximum of e(Gn) if G n⊅K2(r,cn) and Gn⊅ K3 . The third problem is of a slightly different nature. Let Ck(t) be obtained from a cycle Ck by multiplying each vertex by t. We shall prove that if c > 0 then there exists a constant l(c) such that if Gn⊅Ck(t) for k = 3, 5, 2l(c) + 1, then one can omit [cn2] edges from G nso that the obtained graph is bipartite, provided n > n0 (c, t).

AB - The theory of extremal graphs without a fixed set of forbidden subgraphs is well developed. However, rather little is known about extremal graphs without forbidden subgraphs whose orders tend to ∞ with the order of the graph. In this note we deal with three problems of this latter type. Let L be a fixed bipartite graph and let L + Em be the join of L with the empty graph of order m. As our first problem we investigate the maximum of the size e(Gn) of a graph Gn (i.e. a graph of order n) provided Gn⊅L + E[cn, where c > 0 is a constant. In our second problem we study the maximum of e(Gn) if G n⊅K2(r,cn) and Gn⊅ K3 . The third problem is of a slightly different nature. Let Ck(t) be obtained from a cycle Ck by multiplying each vertex by t. We shall prove that if c > 0 then there exists a constant l(c) such that if Gn⊅Ck(t) for k = 3, 5, 2l(c) + 1, then one can omit [cn2] edges from G nso that the obtained graph is bipartite, provided n > n0 (c, t).

UR - http://www.scopus.com/inward/record.url?scp=77957056109&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77957056109&partnerID=8YFLogxK

U2 - 10.1016/S0167-5060(08)70495-3

DO - 10.1016/S0167-5060(08)70495-3

M3 - Article

AN - SCOPUS:77957056109

VL - 3

SP - 29

EP - 41

JO - Annals of Discrete Mathematics

JF - Annals of Discrete Mathematics

SN - 0167-5060

IS - C

ER -