Implicit representation of sparse hereditary families
Abstract
For a hereditary family of graphs , let n denote the set of all members of on n vertices. The speed of is the function f(n)=|n|. An implicit representation of size (n) for n is a function assigning a label of (n) bits to each vertex of any given graph G ∈ n, so that the adjacency between any pair of vertices can be determined by their labels. Bonamy, Esperet, Groenland and Scott proved that the minimum possible size of an implicit representation of n for any hereditary family with speed 2(n2) is (1+o(1)) 2 |n|/n~(=(n)). A recent result of Hatami and Hatami shows that the situation is very different for very sparse hereditary families. They showed that for every δ>0 there are hereditary families of graphs with speed 2O(n n) that do not admit implicit representations of size smaller than n1/2-δ. In this note we show that even a mild speed bound ensures an implicit representation of size O(nc) for some c<1. Specifically we prove that for every >0 there is an integer d ≥ 1 so that if is a hereditary family with speed f(n) ≤ 2(1/4-)n2 then n admits an implicit representation of size O(n1-1/d n). Moreover, for every integer d>1 there is a hereditary family for which this is tight up to the logarithmic factor.