We investigate which graphs H have the property that in every graph with bounded clique number and sufficiently large chromatic number, some induced subgraph is isomorphic to a subdivision of H. In an earlier paper [6], the first author proved that every tree has this property; and in another earlier paper with Maria Chudnovsky [2], we proved that every cycle has this property. Here we give a common generalization. Say a “banana” is the union of a set of paths all with the same ends but otherwise disjoint. We prove that if H is obtained from a tree by replacing each edge by a banana then H has the property mentioned.

中文翻译：

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具有大色数的图的诱导子图。VI。香蕉树

我们研究哪些图H具有以下性质：在每个具有有界集团数和足够大色数的图中，某些诱导子图是H细分的同构。在较早的论文中[6]，第一作者证明了每棵树都具有这种特性。在与玛丽亚·楚德诺夫斯基[2]的另一篇早期论文中，我们证明了每个循环都具有这一特性。这里我们给出一个通用的概括。假设“香蕉”是一组路径的结合，这些路径都具有相同的末端但不相交。我们证明，如果通过用香蕉代替每条边从树上获得H，则H具有上述性质。