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. For the family of all
undirected graphs on n vertices Alstrup, Kaplan, Thorup, and Zwick [STOC
2015] give an induced universal graph with O(2n/2) vertices,
matching a lower bound by Moon [Proc. Glasgow Math. Assoc. 1965].
Let k=⌈D/2⌉. 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(k!k2knk) vertices for the family of graphs with n vertices of maximum degree
D. For constant D, Butler gives a lower bound of
Ω(nD/2). For an odd constant D≥3, Esperet et al.
and Alon and Capalbo [SODA 2008] give a graph with
O(nk−D1) 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=Ω(log3n), the previous best
upper bound was (⌈D/2⌉n)nO(1) due to Adjiashvili and
Rotbart [ICALP 2014]. We give upper and lower bounds showing that the size is
(⌊D/2⌋⌊n/2⌋)2±O~(D). Hence the optimal size is
2O~(D) and our construction is within a factor of
2O~(D) from this. The previous results were
larger by at least a factor of 2Ω(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