A $(a,b)$-coloring of a graph $G$ associates to each vertex a $b$-subset of a set of $a$ colors in such a way that the color-sets of adjacent vertices are disjoint. We define general reduction tools for $(a,b)$-coloring of graphs for $2\le a/b\le 3$. In particular, using necessary and sufficient conditions for the existence of a $(a,b)$-coloring of a path with prescribed color-sets on its end-vertices, 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 for which McDiarmid-Reed's conjecture asserts that they are all $(9,4)$-colorable. Computations on millions of such graphs generated randomly show that our tools allow to find a $(9,4)$-coloring for each of them except for one specific regular shape of graphs (that can be $(9,4)$-colored by an easy ad-hoc process). We thus obtain computational evidence towards the conjecture of McDiarmid\&Reed.

翻译：图$G$的一个$(a,b)$-着色是指为每个顶点赋予一个包含$b$种颜色的子集（颜色集），该子集取自包含$a$种颜色的颜色集合，且相邻顶点的颜色集互不相交。我们针对$2\le a/b\le 3$范围内的图$(a,b)$-着色问题，定义了通用的约简工具。特别地，基于路径在其端点具有指定颜色集时存在$(a,b)$-着色的充要条件，我们提出了更复杂的$(a,b)$-可着色性约简方法。这些工具的实用性通过在三角形格点图中有限的无三角形导出子图上得到验证——McDiarmid-Reed猜想断言这些子图均为$(9,4)$-可着色的。针对随机生成的数百万个此类图的计算表明，除了一种特殊的正则形状图（可通过简单的特设过程实现$(9,4)$-着色）外，我们的工具能为每个图找到$(9,4)$-着色。由此，我们为McDiarmid与Reed的猜想提供了计算证据。