Two-coloring triples such that in each color class every element is missed at least once

TL;DR

This paper characterizes when finite sets of triples can be two-colored so that each element in each color class is missed by at least one triple, providing a linear-time decision algorithm and exploring related generalizations.

Contribution

It offers a characterization and a linear-time algorithm for two-coloring triples with a specific 'missed element' property, extending to hypergraph colorings.

Findings

A linear-time algorithm decides the existence of such a coloring.

Characterization of sets of triples that admit the two-coloring.

Application to hypergraph coloring problems.

Abstract

We give a characterization of finite sets of triples of elements (e.g., positive integers) that can be colored with two colors such that for every element $i$ in each color class there exists a triple which does not contain $i$. We give a linear (in the number of triples) time algorithm to decide if such a coloring exists and find one if it does. We also consider generalizations of this result and an application to a matching problem, which motivated this study. Finally, we show how these results translate to results about colorings of hypergraphs in which the degree of every vertex is $k$ less than the number of hyperedges.