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.

中文翻译：

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1-细分、色数分数和霍尔比

图 G 的霍尔比是 G 的所有子图 H 上 |V(H)|/alpha(H) 的最大值。显然，图的霍尔比是分数色数的下限。有人问是否相反，分数色数的上限是霍尔比的函数。我们通过显示关于 1-细分（图的 1-细分是通过将每条边精确细分一次获得）的两个独立兴趣结果来否定这个问题。* 对于每个 c > 0，每个具有足够大平均度的图都包含至少 c 的分数色数图的 1 细分作为子图。* 对于每一个 d > 0，存在一个平均度数至少为 d 的图 G，使得每个 1-细分出现为 G 的子图的图的霍尔比最多为 18。