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 Molloy（以及 Bernshteyn 的后续工作）最近在无三角形图的（列表）色数上取得的突破性结果的两个明显且自然的改进。在我们的两个结果中，我们允许较低阶数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。我们对第二个问题的证明说明了使用硬核模型来证明约翰逊型结果，这可能具有独立意义。