### Abstract

The Erdos-Gallai Theorem states that for k≥2, every graph of average degree more than k-2 contains a k-vertex path. This result is a consequence of a stronger result of Kopylov: if k is odd, k=2t+1≥5, n≥(5t-3)/2, and G is an n-vertex 2-connected graph with at least h(n,k,t):=(k-t2)+t(n-k+t) edges, then G contains a cycle of length at least k unless G=Hn,k,t:=Kn-E(Kn-t).In this paper we prove a stability version of the Erdos-Gallai Theorem: we show that for all n≥3t>3, and k∈(2t+1,2t+2), every n-vertex 2-connected graph G with e(G)>h(n,k,t-1) either contains a cycle of length at least k or contains a set of t vertices whose removal gives a star forest. In particular, if k=2t+1≠7, we show G⊆Hn,k,t. The lower bound e(G)>h(n,k,t-1) in these results is tight and is smaller than Kopylov's bound h(n,k,t) by a term of n-t-O(1).

Original language | English |
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Journal | Journal of Combinatorial Theory. Series B |

DOIs | |

Publication status | Accepted/In press - Jul 19 2015 |

### Fingerprint

### Keywords

- Cycles
- Paths
- Turán problem

### ASJC Scopus subject areas

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

### Cite this

*Journal of Combinatorial Theory. Series B*. https://doi.org/10.1016/j.jctb.2016.06.004

**Stability in the Erdos-Gallai Theorems on cycles and paths.** / Füredi, Z.; Kostochka, Alexandr; Verstraëte, Jacques.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory. Series B*. https://doi.org/10.1016/j.jctb.2016.06.004

}

TY - JOUR

T1 - Stability in the Erdos-Gallai Theorems on cycles and paths

AU - Füredi, Z.

AU - Kostochka, Alexandr

AU - Verstraëte, Jacques

PY - 2015/7/19

Y1 - 2015/7/19

N2 - The Erdos-Gallai Theorem states that for k≥2, every graph of average degree more than k-2 contains a k-vertex path. This result is a consequence of a stronger result of Kopylov: if k is odd, k=2t+1≥5, n≥(5t-3)/2, and G is an n-vertex 2-connected graph with at least h(n,k,t):=(k-t2)+t(n-k+t) edges, then G contains a cycle of length at least k unless G=Hn,k,t:=Kn-E(Kn-t).In this paper we prove a stability version of the Erdos-Gallai Theorem: we show that for all n≥3t>3, and k∈(2t+1,2t+2), every n-vertex 2-connected graph G with e(G)>h(n,k,t-1) either contains a cycle of length at least k or contains a set of t vertices whose removal gives a star forest. In particular, if k=2t+1≠7, we show G⊆Hn,k,t. The lower bound e(G)>h(n,k,t-1) in these results is tight and is smaller than Kopylov's bound h(n,k,t) by a term of n-t-O(1).

AB - The Erdos-Gallai Theorem states that for k≥2, every graph of average degree more than k-2 contains a k-vertex path. This result is a consequence of a stronger result of Kopylov: if k is odd, k=2t+1≥5, n≥(5t-3)/2, and G is an n-vertex 2-connected graph with at least h(n,k,t):=(k-t2)+t(n-k+t) edges, then G contains a cycle of length at least k unless G=Hn,k,t:=Kn-E(Kn-t).In this paper we prove a stability version of the Erdos-Gallai Theorem: we show that for all n≥3t>3, and k∈(2t+1,2t+2), every n-vertex 2-connected graph G with e(G)>h(n,k,t-1) either contains a cycle of length at least k or contains a set of t vertices whose removal gives a star forest. In particular, if k=2t+1≠7, we show G⊆Hn,k,t. The lower bound e(G)>h(n,k,t-1) in these results is tight and is smaller than Kopylov's bound h(n,k,t) by a term of n-t-O(1).

KW - Cycles

KW - Paths

KW - Turán problem

UR - http://www.scopus.com/inward/record.url?scp=84977642324&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84977642324&partnerID=8YFLogxK

U2 - 10.1016/j.jctb.2016.06.004

DO - 10.1016/j.jctb.2016.06.004

M3 - Article

AN - SCOPUS:84977642324

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

ER -