Fire retainment on Cayley graphs

Gideon Amir, Rangel Baldasso, Maria Gerasimova, Gady Kozma

Research output: Contribution to journalArticlepeer-review


We study the fire-retaining problem on groups, a quasi-isometry invariant1 introduced by Martínez-Pedroza and Prytuła [8], related to the firefighter problem. We prove that any Cayley graph with degree-d polynomial growth does not satisfy {f(n)}-retainment, for any f(n)=o(nd−2), matching the upper bound given for the firefighter problem for these graphs. In the exponential growth regime we prove general lower bounds for direct products and wreath products. These bounds are tight, and show that for exponential-growth groups a wide variety of behaviors is possible. In particular, we construct, for any d≥1, groups that satisfy {nd}-retainment but not o(nd)-retainment, as well as groups that do not satisfy sub-exponential retainment.

Original languageEnglish
Article number113176
JournalDiscrete Mathematics
Issue number1
StatePublished - Jan 2023

Bibliographical note

Funding Information:
Acknowledgments. GA and MG were supported by the Israel Science Foundation Grant 957/20. RB has counted on the support of the Mathematical Institute of Leiden University. GK was supported by the Israel Science Foundation Grant 607/21 and by the Jesselson Foundation.

Publisher Copyright:
© 2022 Elsevier B.V.


  • Cayley graphs
  • Fire containment


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