Academic Journal
Slow graph bootstrap percolation II: Accelerating properties
| Τίτλος: | Slow graph bootstrap percolation II: Accelerating properties |
|---|---|
| Συγγραφείς: | David Fabian, Patrick Morris, Tibor Szabó |
| Συνεισφορές: | Universitat Politècnica de Catalunya. Departament de Matemàtiques |
| Πηγή: | UPCommons. Portal del coneixement obert de la UPC Universitat Politècnica de Catalunya (UPC) |
| Publication Status: | Preprint |
| Στοιχεία εκδότη: | Elsevier BV, 2025. |
| Έτος έκδοσης: | 2025 |
| Θεματικοί όροι: | Running times, Graph bootstrap percolation, Extremal graph theory, FOS: Mathematics, Mathematics - Combinatorics, Combinatorics (math.CO), Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Teoria de grafs |
| Περιγραφή: | 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 $M_H(n)$, and its dependence on properties of the graph $H$. Our focus is on $H$ which define relatively fast bootstrap processes, that is, with $M_H(n)$ being at most linear in $n$. We study the graph class of trees, showing that one can bound $M_T(n)$ 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 $H$. 27 pages, 6 figures. Version updated thanks to comments of referees. To appear in JCTB |
| Τύπος εγγράφου: | Article |
| Περιγραφή αρχείου: | application/pdf |
| Γλώσσα: | English |
| ISSN: | 0095-8956 |
| DOI: | 10.1016/j.jctb.2024.12.006 |
| DOI: | 10.48550/arxiv.2311.18786 |
| Σύνδεσμος πρόσβασης: | http://arxiv.org/abs/2311.18786 |
| Rights: | Elsevier TDM CC BY CC BY NC ND |
| Αριθμός Καταχώρησης: | edsair.doi.dedup.....1a97338fe59d05e00dff7b02941fb18a |
| Βάση Δεδομένων: | OpenAIRE |
| ISSN: | 00958956 |
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| DOI: | 10.1016/j.jctb.2024.12.006 |