Neighbourhood complexity of graphs of bounded twin-width

DOI10.1016/J.EJC.2023.103772zbMATH Open1525.05179arXiv2301.04217OpenAlexW4385473014MaRDI QIDQ6081102

Author name not available (Why is that?)

Publication date: 25 October 2023

Published in: (Search for Journal in Brave)

Abstract: We give essentially tight bounds for, $u(d,k)$, 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., Algorithmica '22; Przybyszewski '22] have shown the upper bound $u(d,k)leqslantexp(exp(O(d)))k$, with a double-exponential dependence in the twin-width. We give a short self-contained proof that for every $d$ and $k$, ${\displaystyle }$ u(d,k) leqslant (d+2)2^{d+1}k = 2^{d+O(log d)}k,${\displaystyle }$ and build a bipartite graph implying $u(d,k)geqslant{2}^{d+logd+O(1)}k$, in the regime when $k$ is large enough compared to $d$.

Full work available at URL: https://arxiv.org/abs/2301.04217

zbMATH Keywords

No records found.

Mathematics Subject Classification ID

No records found.

This page was built for publication: Neighbourhood complexity of graphs of bounded twin-width

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6081102)