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Neighbourhood complexity of graphs of bounded twin-width.
- Source :
-
European Journal of Combinatorics . Jan2024, Vol. 115, pN.PAG-N.PAG. 1p. - Publication Year :
- 2024
-
Abstract
- We give essentially tight bounds for, ν (d , k) , the maximum number of distinct neighbourhoods on a set X of k vertices in a graph with twin-width at most d. Using the celebrated Marcus–Tardos theorem, two independent works (Bonnet et al., 2022; Przybyszewski, 2022) have shown the upper bound ν (d , k) ⩽ exp (exp (O (d))) k , with a double-exponential dependence in the twin-width. The work of Gajarsky et al. (2022), using the framework of local types, implies the existence of a single-exponential bound (without explicitly stating such a bound). We give such an explicit bound, and prove that it is essentially tight. Indeed, we give a short self-contained proof that for every d and k ν (d , k) ⩽ (d + 2) 2 d + 1 k = 2 d + log d + Θ (1) k , and build a bipartite graph implying ν (d , k) ⩾ 2 d + log d + Θ (1) k , in the regime when k is large enough compared to d. [ABSTRACT FROM AUTHOR]
- Subjects :
- *BIPARTITE graphs
*BOUND states
Subjects
Details
- Language :
- English
- ISSN :
- 01956698
- Volume :
- 115
- Database :
- Academic Search Index
- Journal :
- European Journal of Combinatorics
- Publication Type :
- Academic Journal
- Accession number :
- 172846633
- Full Text :
- https://doi.org/10.1016/j.ejc.2023.103772