Semi-Strong Coloring of Intersecting Hypergraphs

TL;DR

This paper investigates the minimum number of colors needed for c-strong coloring of t-intersecting hypergraphs, revealing that for certain parameters, infinite colors are needed, while for others, finite colors suffice, with some open questions remaining.

Contribution

The paper provides new results on the bounds of colors required for c-strong coloring in t-intersecting hypergraphs, clarifying cases where finite or infinite colors are needed.

Findings

No finite colors suffice when t <= c-2.

Finite colors suffice when t >= c, proven via probabilistic methods.

Open questions remain for the case t = c-1 with c > 2.

Abstract

For any c >= 2, a c-strong coloring of the hypergraph G is an assignment of colors to the vertices of G such that for every edge e of G, the vertices of e are colored by at least min{c,|e|} distinct colors. The hypergraph G is t-intersecting if every two edges of G have at least t vertices in common. We ask: for fixed c >= 2 and t >= 1, what is the minimum number of colors that is sufficient to c-strong color any t-intersecting hypergraphs? The purpose of this note is to answer the question for some values of t and c and, more importantly, to describe the settings for which the question is still open. We show that when t <= c-2, no finite number of colors is sufficient to c-strong color all t-intersecting hypergraphs. It is still unknown whether a finite number of colors suffices for the same task when t = c-1 and c > 2. In the last case, when t >= c, we show with a probabilistic argument that a finite number of colors is sufficient to c-strong color all t-intersecting hypergraphs, but a large gap still remains between the best upper and lower bounds on this number.