Gyárfás and Sumner independently conjectured that for every tree T and integer k there is an integer f (k, T ) such that every graph G with χ(G) > f(k, t) contains either K k or an induced copy of T . We prove a `topological´version of the conjecture: for every tree T and integer k there is g(k, T )
Induced subtrees in graphs of large chromatic number
✍ Scribed by A. Gyárfás; E. Szemeredi; Zs. Tuza
- Publisher
- Elsevier Science
- Year
- 1980
- Tongue
- English
- Weight
- 469 KB
- Volume
- 30
- Category
- Article
- ISSN
- 0012-365X
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✦ Synopsis
Our paper proves special eases of the following conjecture: for any fined tr~,e "J~ there exists a natural number f = fiT) ~o that every triangle-free graph of chromatic number ] T) contains T as au induced subgraph. The main ;csult concerns the case when T has radius two.
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It was proved by Hell and Zhu that, if G is a series-parallel graph of girth at least 2 (3k -1)/2 , then χ c (G) ≤ 4k/(2k -1). In this article, we prove that the girth requirement is sharp, i.e., for any k ≥ 2, there is a series-parallel graph G of girth 2 (3k -1)/2 -1 such that χ c (G) > 4k/(2k -1)