# Forskning ved Københavns Universitet - Københavns Universitet

## Near-Optimal Induced Universal Graphs for Bounded Degree Graphs

Publikation: Bidrag til bog/antologi/rapportKonferencebidrag i proceedingsForskningfagfællebedømt

### Dokumenter

A graph U is an induced universal graph for a family F of graphs if every graph in F is a vertex-induced subgraph of U. We give upper and lower bounds for the size of induced universal graphs for the family of graphs with n vertices of maximum degree D. Our new bounds improve several previous results except for the special cases where D is either near-constant or almost n/2. For constant even D Butler [Graphs and Combinatorics 2009] has shown O(n^(D/2)) and recently Alon and Nenadov [SODA 2017] showed the same bound for constant odd D. For constant D Butler also gave a matching lower bound. For generals graphs, which corresponds to D = n, Alon [Geometric and Functional Analysis, to appear] proved the existence of an induced universal graph with (1+o(1)) \cdot 2^((n-1)/2) vertices, leading to a smaller constant than in the previously best known bound of 16 * 2^(n/2) by Alstrup, Kaplan, Thorup, and Zwick [STOC 2015]. In this paper we give the following lower and upper bound of binom(floor(n/2))(floor(D/2)) * n^(-O(1)) and binom(floor(n/2))(floor(D/2)) * 2^(O(sqrt(D log D) * log(n/D))), respectively, where the upper bound is the main contribution. The proof that it is an induced universal graph relies on a randomized argument. We also give a deterministic upper bound of O(n^k / (k-1)!). These upper bounds are the best known when D <= n/2 - tilde-Omega(n^(3/4)) and either D is even and D = omega(1) or D is odd and D = omega(log n/log log n). In this range we improve asymptotically on the previous best known results by Butler [Graphs and Combinatorics 2009], Esperet, Arnaud and Ochem [IPL 2008], Adjiashvili and Rotbart [ICALP 2014], Alon and Nenadov [SODA 2017], and Alon [Geometric and Functional Analysis, to appear].
Originalsprog Engelsk 44th International Colloquium on Automata, Languages, and Programming (ICALP 201 Ioannis Chatzigiannaki, Piotr Indyk, Fabian Kuhn, Anca Muscholl Schloss Dagstuhl - Leibniz-Zentrum für Informatik 2017 1-14 128 978-3-95977-041-5 https://doi.org/10.4230/LIPIcs.ICALP.2017.128 Udgivet - 2017 44th International Colloquium on Automata, Languages, and Programming - Warzawa, PolenVarighed: 10 jul. 2017 → 14 jul. 2017Konferencens nummer: 44

### Konference

Konference 44th International Colloquium on Automata, Languages, and Programming 44 Polen Warzawa 10/07/2017 → 14/07/2017
Navn Leibniz International Proceedings in Informatics (LIPIcs) 80

ID: 196763418