Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. For a positive integer k, a k-colouring of a graph G = (V, E) is a mapping c : V → {1, 2, . . . , k} such that c(u) = c(v) whenever uv ∈ E. The COLOURING problem is to decide, for a given G and k, whether a k-colouring of G exists. If k is fixed (that is, it is not part of the input), we have the decision problem k-COLOURING instead. We survey known results on the computational complexity of COLOURING and k-COLOURING for graph classes that are characterized by one or two forbidden induced subgraphs. We also consider a number of variants: for example, where the problem is to extend a partial colouring, or where lists of permissible colours are given for each vertex. Finally, we also survey results for graph classes defined by some other forbidden pattern.

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The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractWe prove three complexity results on vertex coloring problems restricted to P k -free graphs, i.e., graphs that do not contain a path on k vertices as an induced subgraph. First of all, we show that the pre-coloring extension version of 5-coloring remains NP-complete when restricted to P 6 -free graphs. Recent results of Hoàng et al. imply that this problem is polynomially solvable on P 5 -free graphs. Secondly, we show that the pre-coloring extension version of 3-coloring is polynomially solvable for P 6 -free graphs. This implies a simpler algorithm for checking the 3-colorability of P 6 -free graphs than the algorithm given by Randerath and Schiermeyer. Finally, we prove that 6-coloring is NP-complete for P 7 -free graphs. This problem was known to be polynomially solvable for P 5 -free graphs and NP-complete for P 8 -free graphs, so there remains one open case.

(2014) 'Coloring graphs without short cycles and long induced paths.', Discrete applied mathematics., 167 . pp. 107-120. Further information on publisher's website:http://dx.doi.org/10. 1016/j.dam.2013.12.008 Publisher's copyright statement: NOTICE: this is the author's version of a work that was accepted for publication in Discrete applied mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reected in this document. Changes may have been made to this work since it was submitted for publication. A denitive version was subsequently published in Discrete applied mathematics, 167, 2014, 10.1016/j.dam.2013.12.008 Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. For an integer k ≥ 1, a graph G is k-colorable if there exists a mapping c : VG → {1, . . . , k} such that c(u) = c(v) whenever u and v are two adjacent vertices. For a fixed integer k ≥ 1, the k-COLORING problem is that of testing whether a given graph is k-colorable. The girth of a graph G is the length of a shortest cycle in G. For any fixed g ≥ 4 we determine a lower bound (g), such that every graph with girth at least g and with no induced path on (g) vertices is 3-colorable. We also show that for all fixed integers k, ≥ 1, the k-COLORING problem can be solved in polynomial time for graphs with no induced cycle on four vertices and no induced path on vertices. As a consequence, for all fixed integers k, ≥ 1 and g ≥ 5, the k-COLORING problem can be solved in polynomial time for graphs with girth at least g and with no induced path on vertices. This result is best possible, as we prove the existence of an integer * , such that already 4-COLORING is NP-complete for graphs with girth 4 and with no induced path on * vertices.

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The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Coloring problem is to test whether a graph has a coloring with at most k colors for some integer k. The Precoloring Extension problem is to decide whether a partial k-coloring of a graph can be extended to a k-coloring of the whole graph for some integer k. The List Coloring problem is to decide whether a graph allows a coloring, such that every vertex u receives a color from some given set L(u). By imposing an upper bound on the size of each L(u) we obtain the -List Coloring problem. We first classify the Precoloring Extension problem and the -List Coloring problem for H-free graphs. We then show that 3-List Coloring is NP-complete for n-vertex graphs of minimum degree n − 2, i.e., for complete graphs minus a matching, whereas List Coloring is fixed-parameter tractable for this graph class when parameterized by the number of vertices of degree n − 2. Finally, for a fixed integer k > 0, the List k-Coloring problem is to decide whether a graph allows a coloring, such that every vertex u receives a color from some given set L(u) that must be a subset of {1, . . . , k}. We show that List 4-Coloring is NPcomplete for P6-free graphs, where P6 is the path on six vertices. This completes the classification of List k-Coloring for P6-free graphs.