On dual-ABAB-free and related hypergraphs

TL;DR

This paper studies special classes of hypergraphs defined by forbidden alternating patterns, providing a new matrix-based characterization and exploring properties of their duals, including coloring limitations.

Contribution

It introduces a vertex-ordering-independent characterization of ABA-free hypergraphs and investigates dual hypergraphs, revealing new coloring constraints.

Findings

Dual-ABAB-free hypergraphs are not always 2-colorable.

New matrix characterization of hypergraphs avoids vertex ordering.

Results extend understanding of coloring properties in pattern-restricted hypergraphs.

Abstract

Geometric motivations warranted the study of hypergraphs on ordered vertices that have no pair of hyperedges that induce an alternation of some given length. Such hypergraphs are called ABA-free, ABAB-free and so on. Since then various coloring and other combinatorial results were proved about these families of hypergraphs. We prove a characterization in terms of their incidence matrices which avoids using the ordering of the vertices. Using this characterization, we prove new results about the dual hypergraphs of ABAB-free hypergraphs. In particular, we show that dual-ABAB-free hypergraphs are not always proper $2$-colorable even if we restrict ourselves to hyperedges that are larger than some parameter $m$.