### Abstract

Given two graphs H and G, let H(G) denote the number of subgraphs of G isomorphic to H. We prove that if H is a bipartite graph with a one-factor, then for every triangle-free graph G with n vertices H(G) ≤ H(T_{2}(n)), where T_{2}(n) denotes the complete bipartite graph of n vertices whose colour classes are as equal as possible. We also prove that if K is a complete t-partite graph of m vertices, r > t, n ≥ max(m, r - 1), then there exists a complete (r - 1)-partite graph G* with n vertices such that K(G) ≤ K(G*) holds for every K_{r}-free graph G with n vertices. In particular, in the class of all K_{r}-free graphs with n vertices the complete balanced (r - 1)-partite graph T_{r-1}(n) has the largest number of subgraphs isomorphic to K_{t} (t <r), C_{4}, K_{2,3}. These generalize some theorems of Turán, Erdös and Sauer.

Original language | English |
---|---|

Pages (from-to) | 31-37 |

Number of pages | 7 |

Journal | Graphs and Combinatorics |

Volume | 7 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 1991 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Mathematics(all)

### Cite this

_{r}-free graphs.

*Graphs and Combinatorics*,

*7*(1), 31-37. https://doi.org/10.1007/BF01789461

**On the maximal number of certain subgraphs in K _{r}-free graphs.** / Györi, Ervin; Pach, János; Simonovits, Miklós.

Research output: Contribution to journal › Article

_{r}-free graphs',

*Graphs and Combinatorics*, vol. 7, no. 1, pp. 31-37. https://doi.org/10.1007/BF01789461

_{r}-free graphs. Graphs and Combinatorics. 1991 Mar;7(1):31-37. https://doi.org/10.1007/BF01789461

}

TY - JOUR

T1 - On the maximal number of certain subgraphs in Kr-free graphs

AU - Györi, Ervin

AU - Pach, János

AU - Simonovits, Miklós

PY - 1991/3

Y1 - 1991/3

N2 - Given two graphs H and G, let H(G) denote the number of subgraphs of G isomorphic to H. We prove that if H is a bipartite graph with a one-factor, then for every triangle-free graph G with n vertices H(G) ≤ H(T2(n)), where T2(n) denotes the complete bipartite graph of n vertices whose colour classes are as equal as possible. We also prove that if K is a complete t-partite graph of m vertices, r > t, n ≥ max(m, r - 1), then there exists a complete (r - 1)-partite graph G* with n vertices such that K(G) ≤ K(G*) holds for every Kr-free graph G with n vertices. In particular, in the class of all Kr-free graphs with n vertices the complete balanced (r - 1)-partite graph Tr-1(n) has the largest number of subgraphs isomorphic to Kt (t 4, K2,3. These generalize some theorems of Turán, Erdös and Sauer.

AB - Given two graphs H and G, let H(G) denote the number of subgraphs of G isomorphic to H. We prove that if H is a bipartite graph with a one-factor, then for every triangle-free graph G with n vertices H(G) ≤ H(T2(n)), where T2(n) denotes the complete bipartite graph of n vertices whose colour classes are as equal as possible. We also prove that if K is a complete t-partite graph of m vertices, r > t, n ≥ max(m, r - 1), then there exists a complete (r - 1)-partite graph G* with n vertices such that K(G) ≤ K(G*) holds for every Kr-free graph G with n vertices. In particular, in the class of all Kr-free graphs with n vertices the complete balanced (r - 1)-partite graph Tr-1(n) has the largest number of subgraphs isomorphic to Kt (t 4, K2,3. These generalize some theorems of Turán, Erdös and Sauer.

UR - http://www.scopus.com/inward/record.url?scp=2342483558&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=2342483558&partnerID=8YFLogxK

U2 - 10.1007/BF01789461

DO - 10.1007/BF01789461

M3 - Article

AN - SCOPUS:2342483558

VL - 7

SP - 31

EP - 37

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

SN - 0911-0119

IS - 1

ER -