树和近线稳定集

Tung Nguyen, Alex Scott, Paul Seymour
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摘要

当 $H$ 是森林时,Gy\'arf\'as-Sumner 猜想意味着,每一个没有与 $H$ 同构的诱导子图并且具有有界剪辑数的图 $G$ 都有一个线性大小的稳定集合。我们无法证明这一点,但我们证明了每一个这样的图 $G$ 都有一个大小为 $|G|^{1-o(1)}$ 的稳定集。如果 $H$ 不是前述图,就不需要这样的稳定集。其次,我们证明了当 $H$ 是一个 "多蘑菇 "时,{em is} 存在一个线性大小的稳定集。因此,我们推导出所有的多重房间都满足 Gy\'arf\'as-Sumner 猜想的 "分数着色 "版本。最后,我们讨论了我们的结果在多色环境中的扩展。
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Trees and near-linear stable sets
When $H$ is a forest, the Gy\'arf\'as-Sumner conjecture implies that every graph $G$ with no induced subgraph isomorphic to $H$ and with bounded clique number has a stable set of linear size. We cannot prove that, but we prove that every such graph $G$ has a stable set of size $|G|^{1-o(1)}$. If $H$ is not a forest, there need not be such a stable set. Second, we prove that when $H$ is a ``multibroom'', there {\em is} a stable set of linear size. As a consequence, we deduce that all multibrooms satisfy a ``fractional colouring'' version of the Gy\'arf\'as-Sumner conjecture. Finally, we discuss extensions of our results to the multicolour setting.
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