The $k$-Cut Model in Deterministic and Random Trees

The \(k\)-cut number of rooted graphs was introduced by Cai et al. as a generalization of the classical cutting model by Meir and Moon. In this paper, we show that all moments of the \(k\)-cut number of conditioned Galton-Watson trees converge after proper rescaling, which implies convergence in dis...

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Bibliographic Details
Published inThe Electronic journal of combinatorics Vol. 28; no. 1
Main Authors Berzunza, Gabriel, Cai, Xing Shi, Holmgren, Cecilia
Format Journal Article
LanguageEnglish
Published 29.01.2021
Subjects
PROFILES
NUMBER
CUTTINGS
RANDOM RECORDS
GALTON-WATSON TREES
RANDOM RECURSIVE TREES
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Summary:The \(k\)-cut number of rooted graphs was introduced by Cai et al. as a generalization of the classical cutting model by Meir and Moon. In this paper, we show that all moments of the \(k\)-cut number of conditioned Galton-Watson trees converge after proper rescaling, which implies convergence in distribution to the same limit law regardless of the offspring distribution of the trees. This extends the result of Janson. Using the same method, we also show that the \(k\)-cut number of various random or deterministic trees of logarithmic height converges in probability to a constant after rescaling, such as random split-trees, uniform random recursive trees, and scale-free random trees. 
ISSN:1077-8926
1097-1440
1077-8926
DOI:10.37236/9486