SPARSE PANCYCLIC SUBGRAPHS OF RANDOM GRAPHS

Yahav Alon; Michael Krivelevich

doi:10.1137/23M1598969

SPARSE PANCYCLIC SUBGRAPHS OF RANDOM GRAPHS

Yahav Alon, Michael Krivelevich

Tel Aviv University

Research output: Contribution to journal › Article › peer-review

Abstract

It is known that the complete graph K_n contains a pancyclic subgraph with n + (1 + o(1)) ∙ log₂ n edges, and that there is no pancyclic graph on n vertices with fewer than n + log₂(n - 1) - 1 edges. We show that, with high probability, G(n,p) contains a pancyclic subgraph with n + (1 + o(1)) log₂ n edges for p ≥ p^∗, where p^∗ = (1 + o(1)) ln n/n, which is right above the threshold for pancyclicity.

Original language	English
Pages (from-to)	562-574
Number of pages	13
Journal	SIAM Journal on Discrete Mathematics
Volume	39
Issue number	1
DOIs	https://doi.org/10.1137/23M1598969
State	Published - 2025

Keywords

cycles
pancyclicity
random graph

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1137/23M1598969

Cite this

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title = "SPARSE PANCYCLIC SUBGRAPHS OF RANDOM GRAPHS",

abstract = "It is known that the complete graph Kn contains a pancyclic subgraph with n + (1 + o(1)) ∙ log2 n edges, and that there is no pancyclic graph on n vertices with fewer than n + log2(n - 1) - 1 edges. We show that, with high probability, G(n,p) contains a pancyclic subgraph with n + (1 + o(1)) log2 n edges for p ≥ p∗, where p∗ = (1 + o(1)) ln n/n, which is right above the threshold for pancyclicity.",

keywords = "cycles, pancyclicity, random graph",

author = "Yahav Alon and Michael Krivelevich",

note = "Publisher Copyright: {\textcopyright} 2025 Society for Industrial and Applied Mathematics.",

year = "2025",

doi = "10.1137/23M1598969",

language = "الإنجليزيّة",

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journal = "SIAM Journal on Discrete Mathematics",

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T1 - SPARSE PANCYCLIC SUBGRAPHS OF RANDOM GRAPHS

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PY - 2025

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N2 - It is known that the complete graph Kn contains a pancyclic subgraph with n + (1 + o(1)) ∙ log2 n edges, and that there is no pancyclic graph on n vertices with fewer than n + log2(n - 1) - 1 edges. We show that, with high probability, G(n,p) contains a pancyclic subgraph with n + (1 + o(1)) log2 n edges for p ≥ p∗, where p∗ = (1 + o(1)) ln n/n, which is right above the threshold for pancyclicity.

AB - It is known that the complete graph Kn contains a pancyclic subgraph with n + (1 + o(1)) ∙ log2 n edges, and that there is no pancyclic graph on n vertices with fewer than n + log2(n - 1) - 1 edges. We show that, with high probability, G(n,p) contains a pancyclic subgraph with n + (1 + o(1)) log2 n edges for p ≥ p∗, where p∗ = (1 + o(1)) ln n/n, which is right above the threshold for pancyclicity.

KW - cycles

KW - pancyclicity

KW - random graph

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U2 - 10.1137/23M1598969

DO - 10.1137/23M1598969

M3 - مقالة

SN - 0895-4801

VL - 39

SP - 562

EP - 574

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

IS - 1

ER -

SPARSE PANCYCLIC SUBGRAPHS OF RANDOM GRAPHS

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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