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Free Graphs.', in 29Th International Symposium on Algorithms and Computation (ISAAC 2018) Durham Research Online Deposited in DRO: 11 September 2018 Version of attached le: Accepted Version Peer-review status of attached le: Peer-reviewed Citation for published item: Klimo§sov¡a,T. and Mal¡k,J. and Masa§r¡k,T. and Novotn¡a,J. and Paulusma, D. and Sl¡vov¡a,V. (2018) 'Colouring (Pr+Ps)-free graphs.', in 29th International Symposium on Algorithms and Computation (ISAAC 2018). Dagstuhl, Germany: Schloss Dagstuhl Leibniz-Zentrum f¤urInformatik, 5:1-5:13. Leibniz international proceedings in informatics. (123). Further information on publisher's website: https://doi.org/10.4230/LIPIcs.ISAAC.2018.5 Publisher's copyright statement: c Tereza Klimo§sov¡a,Josef Mal¡k,Tom¡a§sMasa§r¡k,Jana Novotn¡a,Dani¤elPaulusma, Veronika Sl¡vov¡a;licensed under Creative Commons License CC-BY. 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Durham University Library, Stockton Road, Durham DH1 3LY, United Kingdom Tel : +44 (0)191 334 3042 | Fax : +44 (0)191 334 2971 https://dro.dur.ac.uk ∗ † 1 Colouring (Pr + Ps)-Free Graphs 2 Tereza Klimošová 3 Department of Applied Mathematics, Charles University, Prague, Czech Republic 4 [email protected]ff.cuni.cz 5 Josef Malík 6 Czech Technical University in Prague, Czech Republic 7 malikjo1@fit.cvut.cz 8 Tomáš Masařík 9 Department of Applied Mathematics, Charles University, Prague, Czech Republic 10 [email protected]ff.cuni.cz 11 Jana Novotná 12 Department of Applied Mathematics, Charles University, Prague, Czech Republic 13 [email protected]ff.cuni.cz 14 Daniël Paulusma 15 Department of Computer Science, Durham University, Durham, UK 16 [email protected] 17 Veronika Slívová 18 Computer Science Institute of Charles University, Prague, Czech Republic 19 [email protected]ff.cuni.cz 20 Abstract 21 The k-Colouring problem is to decide if the vertices of a graph can be coloured with at most k 22 colours for a fixed integer k such that no two adjacent vertices are coloured alike. If each vertex u 23 must be assigned a colour from a prescribed list L(u) ⊆ {1, . , k}, then we obtain the List k- 24 Colouring problem. A graph G is H-free if G does not contain H as an induced subgraph. 25 We continue an extensive study into the complexity of these two problems for H-free graphs. 26 We prove that List 3-Colouring is polynomial-time solvable for (P2 + P5)-free graphs and 27 for (P3 + P4)-free graphs. Combining our results with known results yields complete complexity 28 classifications of 3-Colouring and List 3-Colouring on H-free graphs for all graphs H up to 29 seven vertices. We also prove that 5-Colouring is NP-complete for (P3 + P5)-free graphs. 30 2012 ACM Subject Classification G.2.2 Graph Theory 31 Keywords and phrases vertex colouring, H-free graph, linear forest 32 Digital Object Identifier 10.4230/LIPIcs.ISAAC.2018.35 ∗ T. Masařík, J. Novotná and V. Slívová were supported by the project GAUK 1277018 and the grant SVV–2017–260452. T. Klimošová was supported by the Center of Excellence – ITI, project P202/12/G061 of GA ČR, by the Center for Foundations of Modern Computer Science (Charles Univ. project UN- CE/SCI/004), and by the project GAUK 1277018. T. Masařík was also partly supported by the Center of Excellence – ITI, project P202/12/G061 of GA ČR. V. Slívová was partly supported by the project 17-09142S of GA ČR and Charles University project PRIMUS/17/SCI/9. D. Paulusma was supported by the Leverhulme Trust (RPG-2016-258). † A preprint of this article is available on arXiv [22]. © Tereza Klimošová, Josef Malík, Tomáš Masařík, Jana Novotná, Daniël Paulusma, Veronika Slívová; licensed under Creative Commons License CC-BY 29th International Symposium on Algorithms and Computation (ISAAC 2018). Editors: Wen-Lian Hsu, Der-Tsai Lee, and Chung-Shou Liao; Article No. 35; pp. 35:1–35:12 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 35:2 Colouring (Pr + Ps)-Free Graphs 33 1 Introduction 34 Graph colouring is a popular concept in Computer Science and Mathematics due to a wide 35 range of practical and theoretical applications, as evidenced by numerous surveys and books 36 on graph colouring and many of its variants (see, for example, [5, 14, 21, 24, 28, 30, 33]). 37 Formally, a colouring of a graph G = (V, E) is a mapping c : V → {1, 2,...} that assigns each 38 vertex u ∈ V a colour c(u) in such a way that c(u) 6= c(v) whenever uv ∈ E. If 1 ≤ c(u) ≤ k, 39 then c is also called a k-colouring of G and G is said to be k-colourable. The Colouring 40 problem is to decide if a given graph G has a k-colouring for some given integer k. 41 It is well known that Colouring is NP-complete even if k = 3 [27]. To pinpoint the 42 reason behind the computational hardness of Colouring one may impose restrictions on the 43 input. This led to an extensive study of Colouring for special graph classes, particularly 44 hereditary graph classes. A graph class is hereditary if it is closed under vertex deletion. 45 As this is a natural property, hereditary graph classes capture a very large collection of 46 well-studied graph classes. It is readily seen that a graph class G is hereditary if and only 47 if G can be characterized by a unique set HG of minimal forbidden induced subgraphs. If 48 HG = {H}, then a graph G ∈ G is called H-free. 49 Král’, Kratochvíl, Tuza, and Woeginger [23] started a systematic study into the complexity 50 of Colouring on H-free graphs for sets H of size at most 2. They showed polynomial- 51 time solvability if H is an induced subgraph of P4 or P1 + P3 and NP-completeness for all 52 other graphs H. The classification for the case where H has size 2 is far from finished; 53 see the summary in [14] or an updated partial overview in [11] for further details. Instead 54 of considering sets H of size 2, we consider H-free graphs and follow another well-studied 55 direction, in which the number of colours k is fixed, that is, k no longer belongs to the input. 56 k-Colouring: Given a graph G does there exist a k-colouring of G? 57 A k-list assignment of G is a function L with domain V such that the list of admissible 58 colours L(u) of each u ∈ V is a subset of {1, 2, . , k}. A colouring c respects L if c(u) ∈ L(u) 59 for every u ∈ V. If k is fixed, then we obtain the following generalization of k-Colouring: List k-Colouring: Given a graph G and a k-list assignment L does there exist a colouring 60 of G that respects L? 61 For every k ≥ 3, k-Colouring on H-free graphs is NP-complete if H contains a cycle [13] 62 or an induced claw [19, 26]. Hence, the case where H is a linear forest (a disjoint union 63 of paths) remains. The situation is far from settled yet, although many partial results are 64 known [2, 3, 4, 6, 7, 8, 9, 10, 15, 18, 20, 25, 29, 31, 34]. Particularly, the case where H is 65 the t-vertex path Pt has been well studied. The cases k = 4, t = 7 and k = 5, t = 6 are 66 NP-complete [20]. For k ≥ 1, t = 5 [18] and k = 3, t = 7 [2], even List k-Colouring 67 on Pt-free graphs is polynomial-time solvable (see also [14]). For a fixed integer k, the 68 k-Precolouring Extension problem is to decide a given k-colouring defined on an induced 69 subgraph of a graph G can be extended to a k-colouring of G. Recently it was shown in [7, 8] 70 that 4-Precolouring Extension, and therefore 4-Colouring, is polynomial-time solvable 71 for P6-free graphs. In contrast, the more general problem List 4-Colouring is NP-complete 72 for P6-free graphs [15]. See Table 1 for a summary of all these results. 73 From Table 1 we see that only the cases k = 3, t ≥ 8 are still open, although some partial 74 results are known for k-Colouring for the case k = 3, t = 8 [9]. The situation when H S 75 is a disconnected linear forest Pi is less clear. It is known that for every s ≥ 1, List 76 3-Colouring is polynomial-time solvable for sP3-free graphs [4, 14]. For every graph H, T. Klimošová, J. Malík, T. Masařík, J. Novotná, D. Paulusma, V. Slívová 35:3 k-Colouring k-Precolouring Extension List k-Colouring t k = 3 k = 4 k = 5 k ≥ 6 k = 3 k = 4 k = 5 k ≥ 6 k = 3 k = 4 k = 5 k ≥ 6 t ≤ 5 P P P P P P P P P P P P t = 6 P P NP-c NP-c P P NP-c NP-c P NP-c NP-c NP-c t = 7 P NP-c NP-c NP-c P NP-c NP-c NP-c P NP-c NP-c NP-c t ≥ 8 ? NP-c NP-c NP-c ? NP-c NP-c NP-c ? NP-c NP-c NP-c Table 1 Summary for Pt-free graphs.
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