We prove two distinct and natural refinements of a recent breakthrough result of Molloy (and a follow‐up work of Bernshteyn) on the (list) chromatic number of triangle‐free graphs. In both our results, we permit the amount of color made available to vertices of lower degree to be accordingly lower. One result concerns list coloring and correspondence coloring, while the other concerns fractional coloring. Our proof of the second illustrates the use of the hard‐core model to prove a Johansson‐type result, which may be of independent interest.

中文翻译：

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使用局部列表大小为无三角形图着色

我们证明了在无三角形图的（列表）色数上，Molloy最近的突破性结果（以及Bernshteyn的后续工作）有两个截然不同的自然改进。在我们的两个结果中，我们允许较低度的顶点可用的颜色量相应地较低。一个结果涉及列表着色和对应着色，而另一个结果涉及分数着色。我们的第二个证明说明了使用硬核模型来证明Johansson型结果，这可能与个人利益有关。