The Burling sequence is a sequence of triangle-free graphs of unbounded chromatic number. The class of Burling graphs consists of all the induced subgraphs of the graphs of this sequence. In the first and second parts of this work, we introduced derived graphs, a class of graphs, equal to the class of Burling graphs, and proved several geometric and structural results about them. In this third part, we use those results to find some Burling and non-Burling graphs, and we see some applications of this in the theory of $\chi$-boundedness. In particular, we show that several graphs, like $ K_5 $, some series-parallel graphs that we call necklaces, and some other graphs are not weakly pervasive.
翻译:键盘序列是无三角的无染色数图的序列。 布尔林图的类别包含此序列图的所有引导子图。 在这项工作的第一和第二部分, 我们引入了衍生图解, 一组图解, 相当于布林图类, 并证明了关于这些图的几何和结构结果 。 在第三部分, 我们用这些结果来找到一些布林图和非布林图, 我们在$\ chi$- burling图的理论中看到这一点的一些应用 。 特别是, 我们展示了一些图表, 比如 K_ 5 美元, 一些我们称之为项链的系列单数图, 还有一些图表并不普遍。