On positive hypergraphs

TL;DR

This paper extends the concept of positive graphs to r-uniform hypergraphs, establishing a key equivalence for odd r and providing an application related to the positivity of subdivided complete bipartite graphs.

Contribution

It introduces the notion of positive hypergraphs and proves their equivalence to positive Levi graphs for odd r, advancing the theoretical understanding of hypergraph positivity.

Findings

Hypergraphs are positive if and only if their Levi graphs are positive for odd r.

The 1-subdivision of K_{r,r} is not positive when r is odd.

Established a new characterization linking hypergraph positivity to Levi graphs.

Abstract

Camarena, Cs\'{o}ka, Hubai, Lippner, and Lov\'{a}sz introduced the notion of positive graphs. This notion naturally extends to $r$-uniform hypergraphs. In the case when $r$ is odd, we prove that a hypergraph is positive if and only if its Levi graph is positive. As an application, we show that the $1$-subdivision of $K_{r,r}$ is not a positive graph when $r$ is odd.