Some mader-perfect graph classes

Rongling Lang, Hui Lei, Siyan Li, Xiaopan Lian, Susu Wang

Abstract: The dichromatic number of D<math><mi is="true">D</mi></math>, denoted by χ→(D)<math><mrow is="true"><mover accent="true" is="true"><mi is="true">χ</mi><mo is="true">→</mo></mover><mrow is="true"><mo is="true">(</mo><mi is="true">D</mi><mo is="true">)</mo></mrow></mrow></math>, is the smallest integer k<math><mi is="true">k</mi></math> such that D<math><mi is="true">D</mi></math> admits an acyclic k<math><mi is="true">k</mi></math>-coloring. We use maderχ→(F)<math><mrow is="true"><msub is="true"><mtext is="true">mader</mtext><mover accent="true" is="true"><mi is="true">χ</mi><mo is="true">→</mo></mover></msub><mrow is="true"><mo is="true">(</mo><mi is="true">F</mi><mo is="true">)</mo></mrow></mrow></math> to denote the smallest integer k<math><mi is="true">k</mi></math> such that if χ→(D)≥k<math><mrow is="true"><mover accent="true" is="true"><mi is="true">χ</mi><mo is="true">→</mo></mover><mrow is="true"><mo is="true">(</mo><mi is="true">D</mi><mo is="true">)</mo></mrow><mo is="true">≥</mo><mi is="true">k</mi></mrow></math>, then D<math><mi is="true">D</mi></math> contains a subdivision of F<math><mi is="true">F</mi></math>. A digraph F<math><mi is="true">F</mi></math> is called Mader-perfect if for every subdigraph F′<math><msup is="true"><mi is="true">F</mi><mo is="true">′</mo></msup></math> of F<math><mi is="true">F</mi></math>, maderχ→(F′)=|V(F′)|<math><mrow is="true"><msub is="true"><mtext is="true">mader</mtext><mover accent="true" is="true"><mi is="true">χ</mi><mo is="true">→</mo></mover></msub><mrow is="true"><mo is="true">(</mo><msup is="true"><mi is="true">F</mi><mo is="true">′</mo></msup><mo is="true">)</mo></mrow><mo linebreak="goodbreak" is="true">=</mo><mrow is="true"><mo is="true">|</mo><mi is="true">V</mi><mrow is="true"><mo is="true">(</mo><msup is="true"><mi is="true">F</mi><mo is="true">′</mo></msup><mo is="true">)</mo></mrow><mo is="true">|</mo></mrow></mrow></math>. We extend octi digraphs to a larger class of digraphs and prove that it is Mader-perfect, which generalizes a result of Gishboliner, Steiner and Szabó [Dichromatic number and forced subdivisions, J. Comb. Theory, Ser. B 153 (2022) 1–30]. We also show that if K<math><mi is="true">K</mi></math> is a proper subdigraph of C↔4<math><msub is="true"><mstyle scriptlevel="0" displaystyle="true" is="true"><mover is="true"><mrow is="true"><mi is="true">C</mi></mrow><mo is="true">↔</mo></mover></mstyle><mn is="true">4</mn></msub></math> except for the digraph obtained from C↔4<math><msub is="true"><mstyle scriptlevel="0" displaystyle="true" is="true"><mover is="true"><mrow is="true"><mi is="true">C</mi></mrow><mo is="true">↔</mo></mover></mstyle><mn is="true">4</mn></msub></math> by deleting an arbitrary arc, then K<math><mi is="true">K</mi></math> is Mader-perfect.