Stability in the Erdos-Gallai Theorems on cycles and paths

Z. Füredi, Alexandr Kostochka, Jacques Verstraëte

Research output: Contribution to journalArticle

3 Citations (Scopus)

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 languageEnglish
JournalJournal of Combinatorial Theory. Series B
DOIs
Publication statusAccepted/In press - Jul 19 2015

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Erdös
Stars
Cycle
Path
Connected graph
Vertex of a graph
Theorem
Star
Odd
Lower bound
Term
Graph in graph theory

Keywords

  • Cycles
  • Paths
  • Turán problem

ASJC Scopus subject areas

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

Cite this

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

In: Journal of Combinatorial Theory. Series B, 19.07.2015.

Research output: Contribution to journalArticle

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