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If each vertex\u00a0<jats:italic>u<\/jats:italic> must be assigned a colour from a prescribed list <jats:inline-formula><jats:alternatives><jats:tex-math>$$L(u)\\subseteq \\{1,\\ldots ,k\\},$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>L<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>u<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                    <mml:mo>\u2286<\/mml:mo>\n                    <mml:mo>{<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mo>\u2026<\/mml:mo>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>k<\/mml:mi>\n                    <mml:mo>}<\/mml:mo>\n                    <mml:mo>,<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> then we obtain the List\u00a0<jats:italic>k<\/jats:italic>-Colouring problem. A graph\u00a0<jats:italic>G<\/jats:italic> is <jats:italic>H<\/jats:italic>-free if <jats:italic>G<\/jats:italic> does not contain <jats:italic>H<\/jats:italic> as an induced subgraph. We continue an extensive study into the complexity of these two problems for <jats:italic>H<\/jats:italic>-free graphs. The graph <jats:inline-formula><jats:alternatives><jats:tex-math>$$P_r+P_s$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>P<\/mml:mi>\n                      <mml:mi>r<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>P<\/mml:mi>\n                      <mml:mi>s<\/mml:mi>\n                    <\/mml:msub>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is the disjoint union of the <jats:italic>r<\/jats:italic>-vertex path\u00a0<jats:inline-formula><jats:alternatives><jats:tex-math>$$P_r$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>P<\/mml:mi>\n                    <mml:mi>r<\/mml:mi>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and the <jats:italic>s<\/jats:italic>-vertex path\u00a0<jats:inline-formula><jats:alternatives><jats:tex-math>$$P_s.$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>P<\/mml:mi>\n                      <mml:mi>s<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mo>.<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> We prove that List\u00a03-Colouring is polynomial-time solvable for <jats:inline-formula><jats:alternatives><jats:tex-math>$$(P_2+P_5)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>P<\/mml:mi>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>P<\/mml:mi>\n                      <mml:mn>5<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-free graphs and for <jats:inline-formula><jats:alternatives><jats:tex-math>$$(P_3+P_4)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>P<\/mml:mi>\n                      <mml:mn>3<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>P<\/mml:mi>\n                      <mml:mn>4<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-free graphs. Combining our results with known results yields complete complexity classifications of 3-Colouring and List\u00a03-Colouring on <jats:italic>H<\/jats:italic>-free graphs for all graphs\u00a0<jats:italic>H<\/jats:italic> up to seven vertices.<\/jats:p>","DOI":"10.1007\/s00453-020-00675-w","type":"journal-article","created":{"date-parts":[[2020,1,25]],"date-time":"2020-01-25T06:02:22Z","timestamp":1579932142000},"page":"1833-1858","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Colouring (Pr\u00a0+\u00a0Ps)-Free Graphs"],"prefix":"10.1007","volume":"82","author":[{"given":"Tereza","family":"Klimo\u0161ov\u00e1","sequence":"first","affiliation":[]},{"given":"Josef","family":"Mal\u00edk","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8524-4036","authenticated-orcid":false,"given":"Tom\u00e1\u0161","family":"Masa\u0159\u00edk","sequence":"additional","affiliation":[]},{"given":"Jana","family":"Novotn\u00e1","sequence":"additional","affiliation":[]},{"given":"Dani\u00ebl","family":"Paulusma","sequence":"additional","affiliation":[]},{"given":"Veronika","family":"Sl\u00edvov\u00e1","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,1,25]]},"reference":[{"key":"675_CR1","series-title":"London Mathematical Society Lecture Note Series","first-page":"1","volume-title":"Surveys in Combinatorics","author":"N Alon","year":"1993","unstructured":"Alon, N.: Restricted colorings of graphs. 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