Clique minors in graphs with a forbidden subgraph

The classical Hadwiger conjecture dating back to 1940's states that any graph of chromatic number at least \(r\) has the clique of order \(r\) as a minor. Hadwiger's conjecture is an example of a well studied class of problems asking how large a clique minor one can guarantee in a graph wi...

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Veröffentlicht in:arXiv.org 2021-02
Hauptverfasser: Bucić, M, Fox, J, Sudakov, B
Format: Artikel
Sprache:eng
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Zusammenfassung:The classical Hadwiger conjecture dating back to 1940's states that any graph of chromatic number at least \(r\) has the clique of order \(r\) as a minor. Hadwiger's conjecture is an example of a well studied class of problems asking how large a clique minor one can guarantee in a graph with certain restrictions. One problem of this type asks what is the largest size of a clique minor in a graph on \(n\) vertices of independence number \(\alpha(G)\) at most \(r\). If true Hadwiger's conjecture would imply the existence of a clique minor of order \(n/\alpha(G)\). Results of Kuhn and Osthus and Krivelevich and Sudakov imply that if one assumes in addition that \(G\) is \(H\)-free for some bipartite graph \(H\) then one can find a polynomially larger clique minor. This has recently been extended to triangle free graphs by Dvořák and Yepremyan, answering a question of Norin. We complete the picture and show that the same is true for arbitrary graph \(H\), answering a question of Dvořák and Yepremyan. In particular, we show that any \(K_s\)-free graph has a clique minor of order \(c_s(n/\alpha(G))^{1+\frac{1}{10(s-2) }}\), for some constant \(c_s\) depending only on \(s\). The exponent in this result is tight up to a constant factor in front of the \(\frac{1}{s-2}\) term.
ISSN:2331-8422