The vertex colourability problem for $$\{claw,butterfly\}$$-free graphs is polynomial-time solvable

Abstract

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.