2012/03/13 by Eric Blais, Blais, Eric, Amit Weinstein +3
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems #Limits and Structures in Graph Theory #math.CO
paper · pdf · doi:10.48550/arxiv.1203.2868
5 pages
arxiv created 2012/03/13 · openalex publication_date 2012/03/13 · arxiv updated 2012/03/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 minc,|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.