Near-Optimal Induced Universal Graphs for Bounded Degree Graphs.

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 $U$. the family of all undirected graphs on $n$ vertices Alstrup, Kaplan, Thorup, and Zwick [STOC 2015] give an induced universal graph with $O!left(2^{n/2}right)$ vertices, matching a lower bound by Moon [Proc. Glasgow Math. Assoc. 1965]. Let $k= lceil D/2 rceil$. Improving asymptotically on previous results by Butler [Graphs and Combinatorics 2009] and Esperet, Arnaud and Ochem [IPL 2008], we give an induced universal graph with $O!left(frac{k2^k}{k!}n^k right)$ vertices for the family of graphs with $n$ vertices of maximum degree $D$. constant $D$, Butler gives a lower bound of $Omega!left(n^{D/2}right)$. an odd constant $Dgeq 3$, Esperet et al. and Alon and Capalbo [SODA 2008] give a graph with $O!left(n^{k-frac{1}{D}}right)$ vertices. Using their techniques for any (including constant) even values of $D$ gives asymptotically worse bounds than we present. For large $D$, i.e. when $D = Omegaleft(log^3 nright)$, the previous best upper bound was ${nchooselceil D/2rceil} n^{O(1)}$ due to Adjiashvili and Rotbart [ICALP 2014]. We give upper and lower bounds showing that the size is ${lfloor n/2rfloorchooselfloor D/2 rfloor}2^{pmtilde{O}left(sqrt{D}right)}$. Hence the optimal size is $2^{tilde{O}(D)}$ and our construction is within a factor of $2^{tilde{O}left(sqrt{D}right)}$ from this. The previous results were larger by at least a factor of $2^{Omega(D)}$. As a part of the above, proving a conjecture by Esperet et al., we construct an induced universal graph with $2n-1$ vertices for the family of graphs with max degree $2$. In addition, we give results for acyclic graphs with max degree $2$ and cycle graphs. Our results imply the first labeling schemes that for any $D$ are at most $o(n)$ bits from optimal.