Colouring random subgraphs

Boris Bukh; Michael Krivelevich; Bhargav Narayanan

doi:10.1017/S0963548325000069

Colouring random subgraphs

Boris Bukh^*, Michael Krivelevich, Bhargav Narayanan

^*Corresponding author for this work

School of Mathematical Sciences

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

Abstract

We study several basic problems about colouring the p-random subgraph G_p of an arbitrary graph G, focusing primarily on the chromatic number and colouring number of G_p. In particular, we show that there exist infinitely many k-regular graphs G for which the colouring number (i.e., degeneracy) of G₁/₂ is at most k/3 + o(k) with high probability, thus disproving the natural prediction that such random graphs must have colouring number at least k/2 − o(k).

Original language	English
Journal	Combinatorics Probability and Computing
DOIs	https://doi.org/10.1017/S0963548325000069
State	Accepted/In press - 2025

Funding

Funders	Funder number
National Science Foundation	DMS-2237138, CCF-1814409, DMS-1555149, DMS-2154063

Keywords

bootstrap percolation
colouring number
Degeneracy

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10.1017/S0963548325000069

Cite this

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abstract = "We study several basic problems about colouring the p-random subgraph Gp of an arbitrary graph G, focusing primarily on the chromatic number and colouring number of Gp. In particular, we show that there exist infinitely many k-regular graphs G for which the colouring number (i.e., degeneracy) of G1/2 is at most k/3 + o(k) with high probability, thus disproving the natural prediction that such random graphs must have colouring number at least k/2 − o(k).",

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AU - Narayanan, Bhargav

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AB - We study several basic problems about colouring the p-random subgraph Gp of an arbitrary graph G, focusing primarily on the chromatic number and colouring number of Gp. In particular, we show that there exist infinitely many k-regular graphs G for which the colouring number (i.e., degeneracy) of G1/2 is at most k/3 + o(k) with high probability, thus disproving the natural prediction that such random graphs must have colouring number at least k/2 − o(k).

KW - bootstrap percolation

KW - colouring number

KW - Degeneracy

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Colouring random subgraphs

Abstract

Funding

Keywords

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