TY - JOUR

T1 - Short proofs for long induced paths

AU - Draganić, Nemanja

AU - Glock, Stefan

AU - Krivelevich, Michael

N1 - Publisher Copyright:
© The Author(s), 2022. Published by Cambridge University Press.

PY - 2022

Y1 - 2022

N2 - 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.

AB - 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.

KW - DFS

KW - induced paths

KW - random graphs

KW - size-Ramsey numbers

KW - supercritical

UR - http://www.scopus.com/inward/record.url?scp=85124951372&partnerID=8YFLogxK

U2 - 10.1017/S0963548322000013

DO - 10.1017/S0963548322000013

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AN - SCOPUS:85124951372

SN - 0963-5483

VL - 31

SP - 870

EP - 878

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

IS - 5

ER -