Short proofs for long induced paths

Nemanja Draganić*, Stefan Glock, Michael Krivelevich

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

We present a modification of the Depth first search algorithm, suited for finding long induced paths. We use it to give simple proofs of the following results. We show that the induced size-Ramsey number of paths satisfies Rind(Pn) ≤ 5.107 n thus giving an explicit constant in the linear bound, improving the previous bound with a large constant from a regularity lemma argument by Haxell, Kohayakawa and Łuczak. We also provide a bound for the k-colour version, showing that Rkind(Pn)=O(k3log4k)n. Finally, we present a new short proof of the fact that the binomial random graph in the supercritical regime, G(n,1ϵ/n), contains typically an induced path of length <![CDATA[$\Theta(ϵ2) n.

Original languageEnglish
Pages (from-to)870-878
Number of pages9
JournalCombinatorics Probability and Computing
Volume31
Issue number5
DOIs
StatePublished - 2022

Funding

FundersFunder number
USA-Israel BSF2018267
Walter Haefner Foundation
Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung200021_196965
Israel Science Foundation1261/17
ETH Zürich Foundation

    Keywords

    • DFS
    • induced paths
    • random graphs
    • size-Ramsey numbers
    • supercritical

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