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The firefighter problem on polynomial and intermediate growth groups
Discrete Mathematics ( IF 0.8 ) Pub Date : 2020-11-01 , DOI: 10.1016/j.disc.2020.112077
Gideon Amir , Rangel Baldasso , Gady Kozma

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.

中文翻译:

多项式和中间增长群的消防员问题

我们证明,对于任何 $f=o(n^{d-2})$,任何具有 $d$ 多项式增长的 Cayley 图 $G$ 都不满足 $\{f(n)\}$-containment。这解决了消防员问题在此类图上的渐近行为,因为已知 $Cn^{d-2}$ 消防员就足够了,回答并加强了 Develin 和 Hartke 的猜想。我们还证明了中间增长 Cayley 图不满足多项式包含,并根据组的增长率给出明确的下界。当有更多几何信息可用时,可以进一步改进这些边界,例如 Grigorchuk 的组。
更新日期:2020-11-01
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