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

TL;DR

This paper establishes a new upper bound on the number of distinct traces in hypergraphs based on degeneracy, leading to improved bounds on VC dimension and applications to graph parameters.

Contribution

It introduces a simple upper bound for trace functions of hypergraphs using degeneracy, generalizing previous results and providing new bounds for VC dimension and related graph parameters.

Findings

Upper bound on traces: at most k * degeneracy of H.

VC dimension of H is at most log(degeneracy)+1.

Reduces known bounds on VC dimension for minor-excluding graphs.

Abstract

Let ${H}=(V, {E})$ be a hypergraph on the vertex set $V$ and edge set ${E}\subseteq 2^V$. We show that number of distinct {\it traces} on any $k-$ subset of $V$, is most $k.{\hat \alpha}(H)$, where ${\hat \alpha}(H)$ is the {\it 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 $\log({\hat \alpha}(H))+1$ which was not known before. As a consequence $vc(H)$ can be computed in computed in $n^{O( {\rm log}({\hat \delta}(H)))}$ 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 $\Omega(n/({\hat \alpha}(G))$, where ${\hat \alpha}(G)$ is the degeneracy of $G$.