Sierpiński products of r-uniform hypergraphs

Abstract

If H1 and H2 are r-uniform hypergraphs and f is a function from the set of all (r − 1)-element subsets of V(H1) into V(H2), then the Sierpinski product H1⊗fH2 is defined to have vertex set V(H1) × V(H2) and hyperedges falling into two classes: (g, h1)(g, h2)⋯(g, hr), such that g ∈ V(H1) and h1h2⋯hr ∈ E(H2),and (g1, f({g2, g3, …, gr}))(g2, f({g1, g3, …, gr}))⋯(gr, f({g1, g2, …, gr − 1})),such that g1g2⋯gr ∈ E(H1). We develop the basic structure possessed by this product and offer proofs of numerous extremal properties involving connectivity, clique numbers, and strong chromatic numbers.