Let %(n, rn) denote the class of simple graphs on n vertices and rn edges and let G E %(n, rn). There are many results in graph theory giving conditions under which G contains certain types of subgraphs, such as cycles of given lengths, complete graphs, etc. For example, Turan's theorem gives a suff
Subgraphs with large degrees and girth
β Scribed by Carsten Thomassen
- Book ID
- 110567490
- Publisher
- Springer Japan
- Year
- 1985
- Tongue
- English
- Weight
- 63 KB
- Volume
- 1
- Category
- Article
- ISSN
- 0911-0119
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## Abstract Let __t__(__n, k__) denote the TurΓ‘n numberβthe maximum number of edges in a graph on __n__ vertices that does not contain a complete graph __K__~k+1~. It is shown that if __G__ is a graph on __n__ vertices with __n__ β₯ __k__^2^(__k__ β 1)/4 and __m__ < __t__(__n, k__) edges, then __G__
In 1983 C. Thomassen conjectured that for every k, g β N there exists d such that any graph with average degree at least d contains a subgraph with average degree at least k and girth at least g. KΓΌhn and Osthus [2004] proved the case g = 6. We give another proof for the case g = 6 which is based