Abstract
The Hall ratio of a graph G is the maximum of |V(H)|/α(H) over all subgraphs H of G. It is easy to see that 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 cl