Colorings at minimum cost

Robert Berke, Dieter Mitsche

Abstract: We define by minc∑{u,v}∈E(G)|c(u)−c(v)|<math><msub is="true"><mrow is="true"><mo is="true">min</mo></mrow><mrow is="true"><mi is="true">c</mi></mrow></msub><msub is="true"><mrow is="true"><mo is="true">∑</mo></mrow><mrow is="true"><mrow is="true"><mo is="true">{</mo><mi is="true">u</mi><mo is="true">,</mo><mi is="true">v</mi><mo is="true">}</mo></mrow><mo is="true">∈</mo><mi is="true">E</mi><mrow is="true"><mo is="true">(</mo><mi is="true">G</mi><mo is="true">)</mo></mrow></mrow></msub><mrow is="true"><mo is="true">|</mo><mi is="true">c</mi><mrow is="true"><mo is="true">(</mo><mi is="true">u</mi><mo is="true">)</mo></mrow><mo is="true">−</mo><mi is="true">c</mi><mrow is="true"><mo is="true">(</mo><mi is="true">v</mi><mo is="true">)</mo></mrow><mo is="true">|</mo></mrow></math> the min-cost MC(G)<math><mi is="true">M</mi><mi is="true">C</mi><mrow is="true"><mo is="true">(</mo><mi is="true">G</mi><mo is="true">)</mo></mrow></math> of a graph G<math><mi is="true">G</mi></math>, where the minimum is taken over all proper colorings c<math><mi is="true">c</mi></math>. The min-cost-chromatic number χM(G)<math><msub is="true"><mrow is="true"><mi is="true">χ</mi></mrow><mrow is="true"><mi is="true">M</mi></mrow></msub><mrow is="true"><mo is="true">(</mo><mi is="true">G</mi><mo is="true">)</mo></mrow></math> is then defined to be the (smallest) number of colors k<math><mi is="true">k</mi></math> for which there exists a proper k<math><mi is="true">k</mi></math>-coloring c<math><mi is="true">c</mi></math> attaining MC(G)<math><mi is="true">M</mi><mi is="true">C</mi><mrow is="true"><mo is="true">(</mo><mi is="true">G</mi><mo is="true">)</mo></mrow></math>. We give constructions of graphs G<math><mi is="true">G</mi></math> where χ(G)<math><mi is="true">χ</mi><mrow is="true"><mo is="true">(</mo><mi is="true">G</mi><mo is="true">)</mo></mrow></math> is arbitrarily smaller than χM(G)<math><msub is="true"><mrow is="true"><mi is="true">χ</mi></mrow><mrow is="true"><mi is="true">M</mi></mrow></msub><mrow is="true"><mo is="true">(</mo><mi is="true">G</mi><mo is="true">)</mo></mrow></math>. On the other hand, we prove that for every 3-regular graph G′<math><msup is="true"><mrow is="true"><mi is="true">G</mi></mrow><mrow is="true"><mo is="true">′</mo></mrow></msup></math>, χM(G′)≤4<math><msub is="true"><mrow is="true"><mi is="true">χ</mi></mrow><mrow is="true"><mi is="true">M</mi></mrow></msub><mrow is="true"><mo is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">G</mi></mrow><mrow is="true"><mo is="true">′</mo></mrow></msup><mo is="true">)</mo></mrow><mo is="true">≤</mo><mn is="true">4</mn></math> and for every 4-regular line graph G″<math><msup is="true"><mrow is="true"><mi is="true">G</mi></mrow><mrow is="true"><mo is="true">″</mo></mrow></msup></math>, χM(G″)≤5<math><msub is="true"><mrow is="true"><mi is="true">χ</mi></mrow><mrow is="true"><mi is="true">M</mi></mrow></msub><mrow is="true"><mo is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">G</mi></mrow><mrow is="true"><mo is="true">″</mo></mrow></msup><mo is="true">)</mo></mrow><mo is="true">≤</mo><mn is="true">5</mn></math>. Moreover, we show that the decision problem whether χM(G)=k<math><msub is="true"><mrow is="true"><mi is="true">χ</mi></mrow><mrow is="true"><mi is="true">M</mi></mrow></msub><mrow is="true"><mo is="true">(</mo><mi is="true">G</mi><mo is="true">)</mo></mrow><mo is="true">=</mo><mi is="true">k</mi></math> is NP<math><mstyle mathvariant="normal" is="true"><mi is="true">NP</mi></mstyle></math>-hard for k≥3<math><mi is="true">k</mi><mo is="true">≥</mo><mn is="true">3</mn></math>.