We present an algorithm to 3-colour a graph G without triangles or induced paths on seven vertices in O(|V (G)| 7 ) time. In fact, our algorithm solves the list 3-colouring problem, where each vertex is assigned a subset of {1, 2, 3} as its admissible colours.

We survey the state of the union-closed sets conjecture.

We prove that for every graph, any vertex subset S, and given integers k,ℓ: there are k disjoint cycles of length at least ℓ that each contain at least one vertex from S, or a vertex set of size O(ℓ·klogk) that meets all such cycles. This generalizes previous results of Fiorini and Herinckx and of Pontecorvi and Wollan. In addition, we describe an algorithm for our main result that runs in O(klogk·s2·false(f(ℓ)·n+mfalse)) time, where s denotes the cardinality of S.

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We describe an algorithm for generating all k-critical H-free graphs, based on a method of Hoàng et al. Using this algorithm, we prove that there are only finitely many 4-critical (P7, C k )-free graphs, for both k = 4 and k = 5. We also show that there are only finitely many 4-critical graphs (P8, C4)free graphs. For each case of these cases we also give the complete lists of critical graphs and vertexcritical graphs. These results generalize previous work by Hell and Huang, and yield certifying algorithms for the 3-colorability problem in the respective classes.Moreover, we prove that for every t, the class of 4-critical planar Pt-free graphs is finite. We also determine all 27 4-critical planar (P7, C6)-free graphs.We also prove that every P11-free graph of girth at least five is 3-colorable, and show that this is best possible by determining the smallest 4-chromatic P12-free graph of girth at least five. Moreover, we show that every P14-free graph of girth at least six and every P17-free graph of girth at least seven is 3-colorable. This strengthens results of Golovach et al.