关于 Gyárfás-Sumner 猜想的说明

Pub Date : 2024-03-08 DOI:10.1007/s00373-024-02754-z

Tung Nguyen, Alex Scott, Paul Seymour

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引用次数: 0

摘要

Gyárfás-Sumner 猜想说，对于每棵树 T 和每个整数 \（t\ge 1\），如果 G 是一个没有大小为 t 的簇且色度数足够大的图，那么 G 包含一个与 T 同构的诱导子图。这一点仍未解决，但我们证明，在同样的假设下，G 包含一个与 T 同构的子图 H，它是 "路径诱导 "的；也就是说，对于某个区分顶点 r，H 的每条路径的一个端点 r 都是 G 的一条诱导路径。

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A Note on the Gyárfás–Sumner Conjecture

The Gyárfás–Sumner conjecture says that for every tree T and every integer \(t\ge 1\), if G is a graph with no clique of size t and with sufficiently large chromatic number, then G contains an induced subgraph isomorphic to T. This remains open, but we prove that under the same hypotheses, G contains a subgraph H isomorphic to T that is “path-induced”; that is, for some distinguished vertex r, every path of H with one end r is an induced path of G.

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