A $(a,b)$-coloring of a graph $G$ associates to each vertex a set of $b$ colors from a set of $a$ colors in such a way that the color-sets of adjacent vertices are disjoints. We define general reduction tools for $(a,b)$-coloring of graphs for $2\le a/b\le 3$. In particular, we prove necessary and sufficient conditions for the existence of a $(a,b)$-coloring of a path with prescribed color-sets on its end-vertices. Other more complex $(a,b)$-colorability reductions are presented. The utility of these tools is exemplified on finite triangle-free induced subgraphs of the triangular lattice. Computations on millions of such graphs generated randomly show that our tools allow to find (in linear time) a $(9,4)$-coloring for each of them. Although there remain few graphs for which our tools are not sufficient for finding a $(9,4)$-coloring, we believe that pursuing our method can lead to a solution of the conjecture of McDiarmid-Reed.

中文翻译：

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图多色约简方法及其在 McDiarmid-Reed 猜想中的应用

图 $G$ 的 $(a,b)$ 着色将一组 $a$ 颜色中的一组 $b$ 颜色与每个顶点相关联，这样相邻顶点的颜色集是不相交的。我们为$2\le a/b\le 3$ 的图的$(a,b)$-着色定义了通用的缩减工具。特别是，我们证明了存在 $(a,b)$-着色的必要和充分条件，该路径在其末端顶点上具有指定的颜色集。呈现了其他更复杂的 $(a,b)$-可着色性降低。这些工具的效用在三角形晶格的有限三角形诱导子图上得到了例证。对随机生成的数百万个此类图的计算表明，我们的工具允许（在线性时间内）为它们中的每一个找到 $(9,4)$-着色。尽管我们的工具不足以找到 $(9,4)$-着色的图，但仍然很少有图，