Implicit representation of sparse hereditary families
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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 ℓ(n) for _n is a function assigning a label of ℓ(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^Ω(n^2) is (1+o(1)) log_2 |_n|/n (=Θ(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 2^O(n log n) that do not admit implicit representations of size smaller than n^1/2-δ. In this note we show that even a mild speed bound ensures an implicit representation of size O(n^c) 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-)n^2 then _n admits an implicit representation of size O(n^1-1/dlog n). Moreover, for every integer d>1 there is a hereditary family for which this is tight up to the logarithmic factor.
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