Concentration of maximum degree in random planar graphs

M. Kang; M. Missethan

doi:10.1016/j.jctb.2022.05.005

Concentration of maximum degree in random planar graphs

M. Kang, M. Missethan^*

^*Corresponding author for this work

Institute of Discrete Mathematics (5050)

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

Abstract

Let P (n, m) be a graph chosen uniformly at random from the class of all planar graphs on vertex set [n] := {1, . . . , n} with m = m(n) edges. We show that in the sparse regime, when m/n ≤ 1, with high probability the maximum degree of
P (n, m) takes at most two different values. In contrast, this is not true anymore in the dense regime, when m/n > 1, where the maximum degree of P (n, m) is not concentrated on any subset of [n] with bounded size

Original language	English
Pages (from-to)	310-342
Number of pages	33
Journal	Journal of Combinatorial Theory, Series B
Volume	156
DOIs	https://doi.org/10.1016/j.jctb.2022.05.005
Publication status	Published - 2022

Keywords

Balls into bins
Maximum degree
Prüfer sequence
Random graphs
Random planar graphs

ASJC Scopus subject areas

Theoretical Computer Science
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

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10.1016/j.jctb.2022.05.005Licence: CC BY 4.0

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title = "Concentration of maximum degree in random planar graphs",

abstract = "Let P (n, m) be a graph chosen uniformly at random from the class of all planar graphs on vertex set [n] := {1, . . . , n} with m = m(n) edges. We show that in the sparse regime, when m/n ≤ 1, with high probability the maximum degree ofP (n, m) takes at most two different values. In contrast, this is not true anymore in the dense regime, when m/n > 1, where the maximum degree of P (n, m) is not concentrated on any subset of [n] with bounded size",

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author = "M. Kang and M. Missethan",

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language = "English",

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T1 - Concentration of maximum degree in random planar graphs

AU - Kang, M.

AU - Missethan, M.

PY - 2022

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N2 - Let P (n, m) be a graph chosen uniformly at random from the class of all planar graphs on vertex set [n] := {1, . . . , n} with m = m(n) edges. We show that in the sparse regime, when m/n ≤ 1, with high probability the maximum degree ofP (n, m) takes at most two different values. In contrast, this is not true anymore in the dense regime, when m/n > 1, where the maximum degree of P (n, m) is not concentrated on any subset of [n] with bounded size

AB - Let P (n, m) be a graph chosen uniformly at random from the class of all planar graphs on vertex set [n] := {1, . . . , n} with m = m(n) edges. We show that in the sparse regime, when m/n ≤ 1, with high probability the maximum degree ofP (n, m) takes at most two different values. In contrast, this is not true anymore in the dense regime, when m/n > 1, where the maximum degree of P (n, m) is not concentrated on any subset of [n] with bounded size

KW - Balls into bins

KW - Maximum degree

KW - Prüfer sequence

KW - Random graphs

KW - Random planar graphs

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DO - 10.1016/j.jctb.2022.05.005

M3 - Article

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VL - 156

SP - 310

EP - 342

JO - Journal of Combinatorial Theory, Series B

JF - Journal of Combinatorial Theory, Series B

ER -

Concentration of maximum degree in random planar graphs

Abstract

Keywords

ASJC Scopus subject areas

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