Summary: EL%VIER Discrete Mathematics 1651166 (1997) 2 I-30
The complexity of G-free colourability
Demetrios Achlioptas *
Lkpurtmentqf`Computer Science, (/nicer&y of Toronto, 10 Kinq's College Rd
Toronto. Ont., Canada MSS 3G4
The problem of determining if a graph is 2-colourable (i.e., bipartite) has long been known to
have a simple polynomial time algorithm. Being 2-colourable is equivalent to having a bipartition
of the vertex set where each cell is &-free. We extend this notion to determining if there exists
a bipartition where each cell is G-free for some fixed graph G. One might expect that for some
graphs other than K2, K2 there also exist polynomial time algorithms. Rather surprisingly WC
show that for UZ~ graph G on more than two vertices the problem is NP-complete.
A vertex k-colouring of a graph is an assignment of one of k colours to each vertex
such that adjacent vertices receive different colours. Such colourings have been studied
extensively and form one of the oldest and deepest areas of graph theory. In this
course of study many generalisations of the colouring concept have been suggested.
The following two notions, introduced in , appear to be useful in expressing such