A simple upper bound for trace function of a hypergraph with applications

02/22/2019

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by   Farhad Shahrokhi, et al.

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Let H=(V, E) be a hypergraph on the vertex set V and edge set E⊆ 2^V. We show that number of distinct traces on any k- subset of V, is most k.α̂(H), where α̂(H) is the degeneracy of H. The result significantly improves/generalizes some of related results. For instance, the vc dimension H (or vc(H)) is shown to be at most (α̂(H))+1 which was not known before. As a consequence vc(H) can be computed in computed in n^O( log(δ̂(H))) time. When applied to the neighborhood systems of a graphs excluding a fixed minor, it reduces the known linear upper bound on the VC dimension to a logarithmic one, in the size of the minor. When applied to the location domination and identifying code numbers of any n vertex graph G, one gets the new lower bound of Ω(n/(α̂(G)), where α̂(G) is the degeneracy of G.