2018/12/18 by Zdeněk Dvořák, Zdenĕk Dvořák, Patrice Ossona de Mendez +4
Computer Science · Mathematics · #05C15 #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #G.2.2 #Graph Labeling and Dimension Problems #Limits and Structures in Graph Theory #acm:05C15 #math.CO #msc:05C15
paper · pdf · doi:10.48550/arxiv.1812.07327
14 pages, no figures; updated for reviewer remarks
openalex publication_date 2018/12/18 · openalex created_date 2018/12/22 · arxiv created 2020/01/30 · arxiv updated 2020/01/31 · openalex updated_date 2026/07/28
The Hall ratio of a graph G is the maximum of |V(H)|/alpha(H) over all subgraphs H of G. Clearly, the Hall ratio of a graph is a lower bound for the fractional chromatic number. It has been asked whether conversely, the fractional chromatic number is upper bounded by a function of the Hall ratio. We answer this question in negative, by showing two results of independent interest regarding 1-subdivisions (the 1-subdivision of a graph is obtained by subdividing each edge exactly once). * For every c > 0, every graph of sufficiently large average degree contains as a subgraph the 1-subdivision of a graph of fractional chromatic number at least c. * For every d > 0, there exists a graph G of average degree at least d such that every graph whose 1-subdivision appears as a subgraph of G has Hall ratio at most 18. We also discuss the consequences of these results in the context of graph classes with bounded expansion.