The vertex colourability problem is to determine, for a given graph and a given natural k, whether it is possible to split the graph’s vertex set into at most k subsets, each of pairwise non-adjacent vertices, or not. A hereditary class is a set of simple graphs, closed under deletion of vertices. Any such a class can be defined by the set of its forbidden induced subgraphs. For all but four hereditary classes, defined by a pair of connected five-vertex induced prohibitions, the complexity status of the vertex colourability problem is known. In this paper, we reduce the number of the open cases to three, by showing polynomial-time solvability of the problem for \(\{claw,butterfly\}\)-free graphs. A claw is the star graph with three leaves, a butterfly is obtained by coinciding a vertex in a triangle with a vertex in another triangle.

顶点可着色性问题是，对于给定的图和给定的自然k，确定是否有可能将图的顶点集拆分为最多k个子集，每个子​​集都是成对的非相邻顶点。世袭类是一组简单的图，在删除顶点后关闭。任何此类都可以通过其禁止的诱导子图的集合来定义。对于除四个遗传类别（由一对相连的五顶点诱发的禁止所定义）之外的所有其他顶点，已知顶点着色性问题的复杂性状态。在本文中，通过显示无\（\ {claw，butterfly \} \） -图的问题的多项式时间可解性，我们将未处理案例的数量减少到三个。一爪如果是具有三片叶子的星形图，则通过将三角形中的一个顶点与另一个三角形中的一个顶点重合来获得蝴蝶。