Induced C4-free subgraphs with large average degree

Xiying Du, António Girão, Zach Hunter, Rose McCarty, Alex D. Scott

Abstract: We prove that there exists a constant C so that, for all s,k∈N<math><mi is="true">s</mi><mo is="true">,</mo><mi is="true">k</mi><mo is="true">∈</mo><mi mathvariant="double-struck" is="true">N</mi></math>, if G has average degree at least kCs3<math><msup is="true"><mrow is="true"><mi is="true">k</mi></mrow><mrow is="true"><mi is="true">C</mi><msup is="true"><mrow is="true"><mi is="true">s</mi></mrow><mrow is="true"><mn is="true">3</mn></mrow></msup></mrow></msup></math> and does not contain Ks,s<math><msub is="true"><mrow is="true"><mi is="true">K</mi></mrow><mrow is="true"><mi is="true">s</mi><mo is="true">,</mo><mi is="true">s</mi></mrow></msub></math> as a subgraph then it contains an induced subgraph which is C4<math><msub is="true"><mrow is="true"><mi is="true">C</mi></mrow><mrow is="true"><mn is="true">4</mn></mrow></msub></math>-free and has average degree at least k. It was known that some function of s and k suffices, but this is the first explicit bound. We give several applications of this result, including short and streamlined proofs of the following two corollaries.We show that there exists a constant C so that, for all s,k∈N<math><mi is="true">s</mi><mo is="true">,</mo><mi is="true">k</mi><mo is="true">∈</mo><mi mathvariant="double-struck" is="true">N</mi></math>, if G has average degree at least kCs3<math><msup is="true"><mrow is="true"><mi is="true">k</mi></mrow><mrow is="true"><mi is="true">C</mi><msup is="true"><mrow is="true"><mi is="true">s</mi></mrow><mrow is="true"><mn is="true">3</mn></mrow></msup></mrow></msup></math> and does not contain Ks,s<math><msub is="true"><mrow is="true"><mi is="true">K</mi></mrow><mrow is="true"><mi is="true">s</mi><mo is="true">,</mo><mi is="true">s</mi></mrow></msub></math> as a subgraph then it contains an induced subdivision of Kk<math><msub is="true"><mrow is="true"><mi is="true">K</mi></mrow><mrow is="true"><mi is="true">k</mi></mrow></msub></math>. This is the first quantitative improvement on a well-known theorem of Kühn and Osthus; their proof gives a bound that is triply exponential in both k and s.We also show that for any hereditary degree-bounded class F<math><mi mathvariant="script" is="true">F</mi></math>, there exists a constant C=CF<math><mi is="true">C</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><msub is="true"><mrow is="true"><mi is="true">C</mi></mrow><mrow is="true"><mi mathvariant="script" is="true">F</mi></mrow></msub></math> so that Cs3<math><msup is="true"><mrow is="true"><mi is="true">C</mi></mrow><mrow is="true"><msup is="true"><mrow is="true"><mi is="true">s</mi></mrow><mrow is="true"><mn is="true">3</mn></mrow></msup></mrow></msup></math> is a degree-bounding function for F<math><mi mathvariant="script" is="true">F</mi></math>. This is the first bound of any type on the rate of growth of such functions.