GyΓ‘rfΓ‘s 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 Ο(G) > f(k, t) contains either K k or an induced copy of T . We prove a `topologicalΒ΄version of the conjecture: for every tree T and integer k there is g(k, T )
Subgraphs of large connectivity and chromatic number in graphs of large chromatic number
β Scribed by N. Alon; D. Kleitman; C. Thomassen; M. Saks; P. Seymour
- Publisher
- John Wiley and Sons
- Year
- 1987
- Tongue
- English
- Weight
- 144 KB
- Volume
- 11
- Category
- Article
- ISSN
- 0364-9024
No coin nor oath required. For personal study only.
β¦ Synopsis
For each pair k, rn of natural numbers there exists a natural number f(k, rn) such that every f ( k , m)-chromatic graph contains a k-connected subgraph of chromatic number at least rn.
π SIMILAR VOLUMES
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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 Ο c (G) β€ 4k/(2k -1). In this article, we prove that the girth requirement is sharp, i.e., for any k β₯ 2, there is a series-parallel graph G of girth 2 (3k -1)/2 -1 such that Ο c (G) > 4k/(2k -1)