Abstract
A graph U is an induced universal graph for a family F of graphs if every graph in F is a vertexinduced 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 nearconstant 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^((n1)/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 / (k1)!). These upper bounds are the best known when D <= n/2  tildeOmega(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].
BibTeX  Entry
@InProceedings{abrahamsen_et_al:LIPIcs:2017:7411,
author = {Mikkel Abrahamsen and Stephen Alstrup and Jacob Holm and Mathias Bæk Tejs Knudsen and Morten St{\"o}ckel},
title = {{NearOptimal Induced Universal Graphs for Bounded Degree Graphs}},
booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
pages = {128:1128:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770415},
ISSN = {18688969},
year = {2017},
volume = {80},
editor = {Ioannis Chatzigiannakis and Piotr Indyk and Fabian Kuhn and Anca Muscholl},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7411},
URN = {urn:nbn:de:0030drops74114},
doi = {10.4230/LIPIcs.ICALP.2017.128},
annote = {Keywords: Adjacency labeling schemes, Bounded degree graphs, Induced universal graphs, Distributed computing}
}
Keywords: 

Adjacency labeling schemes, Bounded degree graphs, Induced universal graphs, Distributed computing 
Collection: 

44th International Colloquium on Automata, Languages, and Programming (ICALP 2017) 
Issue Date: 

2017 
Date of publication: 

07.07.2017 