### Abstract

The Erdős–Gallai Theorem states that for k≥3, any n-vertex graph with no cycle of length at least k has at most [Formula preseted](k−1)(n−1) edges. A stronger version of the Erdős–Gallai Theorem was given by Kopylov: If G is a 2-connected n-vertex graph with no cycle of length at least k, then e(G)≤max{h(n,k,2),h(n,k,⌊[Formula preseted]⌋)}, where h(n,k,a)≔[Formula preseted]+a(n−k+a). Furthermore, Kopylov presented the two possible extremal graphs, one with h(n,k,2) edges and one with h(n,k,⌊[Formula preseted]⌋) edges. In this paper, we complete a stability theorem which strengthens Kopylov's result. In particular, we show that for k≥3 odd and all n≥k, every n-vertex 2-connected graph G with no cycle of length at least k is a subgraph of one of the two extremal graphs or e(G)≤max{h(n,k,3),h(n,k,[Formula preseted])}. The upper bound for e(G) here is tight.

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

Pages (from-to) | 1253-1263 |

Number of pages | 11 |

Journal | Discrete Mathematics |

Volume | 341 |

Issue number | 5 |

DOIs | |

Publication status | Published - May 1 2018 |

### Fingerprint

### Keywords

- Cycles
- Paths
- Turán problem

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

### Cite this

*Discrete Mathematics*,

*341*(5), 1253-1263. https://doi.org/10.1016/j.disc.2017.12.018

**Stability in the Erdős–Gallai Theorem on cycles and paths, II.** / Füredi, Z.; Kostochka, Alexandr; Luo, Ruth; Verstraëte, Jacques.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 341, no. 5, pp. 1253-1263. https://doi.org/10.1016/j.disc.2017.12.018

}

TY - JOUR

T1 - Stability in the Erdős–Gallai Theorem on cycles and paths, II

AU - Füredi, Z.

AU - Kostochka, Alexandr

AU - Luo, Ruth

AU - Verstraëte, Jacques

PY - 2018/5/1

Y1 - 2018/5/1

N2 - The Erdős–Gallai Theorem states that for k≥3, any n-vertex graph with no cycle of length at least k has at most [Formula preseted](k−1)(n−1) edges. A stronger version of the Erdős–Gallai Theorem was given by Kopylov: If G is a 2-connected n-vertex graph with no cycle of length at least k, then e(G)≤max{h(n,k,2),h(n,k,⌊[Formula preseted]⌋)}, where h(n,k,a)≔[Formula preseted]+a(n−k+a). Furthermore, Kopylov presented the two possible extremal graphs, one with h(n,k,2) edges and one with h(n,k,⌊[Formula preseted]⌋) edges. In this paper, we complete a stability theorem which strengthens Kopylov's result. In particular, we show that for k≥3 odd and all n≥k, every n-vertex 2-connected graph G with no cycle of length at least k is a subgraph of one of the two extremal graphs or e(G)≤max{h(n,k,3),h(n,k,[Formula preseted])}. The upper bound for e(G) here is tight.

AB - The Erdős–Gallai Theorem states that for k≥3, any n-vertex graph with no cycle of length at least k has at most [Formula preseted](k−1)(n−1) edges. A stronger version of the Erdős–Gallai Theorem was given by Kopylov: If G is a 2-connected n-vertex graph with no cycle of length at least k, then e(G)≤max{h(n,k,2),h(n,k,⌊[Formula preseted]⌋)}, where h(n,k,a)≔[Formula preseted]+a(n−k+a). Furthermore, Kopylov presented the two possible extremal graphs, one with h(n,k,2) edges and one with h(n,k,⌊[Formula preseted]⌋) edges. In this paper, we complete a stability theorem which strengthens Kopylov's result. In particular, we show that for k≥3 odd and all n≥k, every n-vertex 2-connected graph G with no cycle of length at least k is a subgraph of one of the two extremal graphs or e(G)≤max{h(n,k,3),h(n,k,[Formula preseted])}. The upper bound for e(G) here is tight.

KW - Cycles

KW - Paths

KW - Turán problem

UR - http://www.scopus.com/inward/record.url?scp=85042255197&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85042255197&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2017.12.018

DO - 10.1016/j.disc.2017.12.018

M3 - Article

AN - SCOPUS:85042255197

VL - 341

SP - 1253

EP - 1263

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 5

ER -