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

Z. Füredi, Alexandr Kostochka, Ruth Luo, Jacques Verstraëte

Research output: Contribution to journalArticle

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 languageEnglish
Pages (from-to)1253-1263
Number of pages11
JournalDiscrete Mathematics
Volume341
Issue number5
DOIs
Publication statusPublished - May 1 2018

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Cycle
Path
Theorem
Extremal Graphs
Vertex of a graph
Stability Theorem
Graph in graph theory
Connected graph
Subgraph
Odd
Upper bound

Keywords

  • Cycles
  • Paths
  • Turán problem

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

Cite this

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

In: Discrete Mathematics, Vol. 341, No. 5, 01.05.2018, p. 1253-1263.

Research output: Contribution to journalArticle

Füredi, Z. ; Kostochka, Alexandr ; Luo, Ruth ; Verstraëte, Jacques. / Stability in the Erdős–Gallai Theorem on cycles and paths, II. In: Discrete Mathematics. 2018 ; Vol. 341, No. 5. pp. 1253-1263.
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