The firefighter problem on polynomial and intermediate growth groups

Gideon Amir; Rangel Baldasso; Gady Kozma

doi:10.1016/j.disc.2020.112077

The firefighter problem on polynomial and intermediate growth groups

Gideon Amir, Rangel Baldasso, Gady Kozma

Department of Mathematics

Weizmann Institute of Science

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

6 Scopus citations

Abstract

We prove that any Cayley graph G with degree d polynomial growth does not satisfy {f(n)}-containment for any f=o(n^d−2). This settles the asymptotic behaviour of the firefighter problem on such graphs as it was known that Cn^d−2 firefighters are enough, answering and strengthening a conjecture of Develin and Hartke. We also prove that intermediate growth Cayley graphs do not satisfy polynomial containment, and give explicit lower bounds depending on the growth rate of the group. These bounds can be further improved when more geometric information is available, such as for Grigorchuk's group.

Original language	English
Article number	112077
Journal	Discrete Mathematics
Volume	343
Issue number	11
DOIs	https://doi.org/10.1016/j.disc.2020.112077
State	Published - Nov 2020

Bibliographical note

Publisher Copyright:
© 2020 Elsevier B.V.

Funding

The authors would like to thank all the suggestions of the anonymous referee. While performing this research, G.A. and R.B. were supported by the Israel Science Foundation grant #575/16 and by GIF grant #I-1363-304.6/2016 . G.K. was supported by the Israel Science Foundation grant #1369/15 and by the Jesselson Foundation .

Funders	Funder number
Jesselson Foundation
German-Israeli Foundation for Scientific Research and Development	1369/15, -1363-304.6/2016
Israel Science Foundation	575/16

Keywords

Cayley graphs
Firefighter problem

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10.1016/j.disc.2020.112077

Cite this

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The firefighter problem on polynomial and intermediate growth groups

Abstract

Bibliographical note

Funding

Keywords

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