Slow graph bootstrap percolation II: Accelerating properties

IF 1.2 1区 数学 Q1 MATHEMATICS
David Fabian , Patrick Morris , Tibor Szabó
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引用次数: 0

Abstract

For a graph H and an n-vertex graph G, the H-bootstrap process on G is the process which starts with G and, at every time step, adds any missing edges on the vertices of G that complete a copy of H. This process eventually stabilises and we are interested in the extremal question raised by Bollobás of determining the maximum running time (number of time steps before stabilising) of this process over all possible choices of n-vertex graph G. In this paper, we initiate a systematic study of the asymptotics of this parameter, denoted MH(n), and its dependence on properties of the graph H. Our focus is on H which define relatively fast bootstrap processes, that is, with MH(n) being at most linear in n. We study the graph class of trees, showing that one can bound MT(n) by a quadratic function in v(T) for all trees T and all n. We then go on to explore the relationship between the running time of the H-process and the minimum vertex degree and connectivity of H.
慢图自举渗透II:加速特性
图H和n点图G, G的H-bootstrap过程始于G和的过程,在每一个时间步,添加任何丢失边缘G的顶点,最终完成一份H .这个过程稳定和我们感兴趣的是极值问题提出Bollobas确定的最大运行时间(稳定前的时间步数)这个过程的所有可能的选择的n点图G .摘要我们开始系统地研究这个参数的渐近性,表示为MH(n),以及它对图H的性质的依赖。我们的重点是H,它定义了相对快速的自举过程,即MH(n)在n中最多是线性的。我们研究树的图类,表明可以通过v(T)中的二次函数对所有树T和所有树n进行约束MT(n)。然后我们继续探索H过程的运行时间与H的最小顶点度和连通性之间的关系。
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来源期刊
CiteScore
2.70
自引率
14.30%
发文量
99
审稿时长
6-12 weeks
期刊介绍: The Journal of Combinatorial Theory publishes original mathematical research dealing with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series B is concerned primarily with graph theory and matroid theory and is a valuable tool for mathematicians and computer scientists.
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