Let Ex(n, k, Β΅) denote the maximum number of edges of an n-vertex graph in which every subgraph of k vertices has at most Β΅ edges. Here we summarize some known results of the problem of determining Ex(n, k, Β΅), give simple proofs, and find some new estimates and extremal graphs. Besides proving new
Decomposing large graphs with small graphs of high density
β Scribed by Yuster, Raphael
- Publisher
- John Wiley and Sons
- Year
- 1999
- Tongue
- English
- Weight
- 146 KB
- Volume
- 32
- Category
- Article
- ISSN
- 0364-9024
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β¦ Synopsis
It is shown that for every positive integer h, and for every > 0, there are graphs H = (V H , E H ) with at least h vertices and with density at least 0.5with the following property:
any graph with minimum degree at least |V G | 2 (1 + o(1)) and |E H | divides |E G |, then G has an H-decomposition. This result extends the results of [
π SIMILAR VOLUMES
It is proved that a planar graph with maximum degree β β₯ 11 has total (vertex-edge) chromatic number β + 1.
The odd-girth of a graph is the length of a shortest odd circuit. A conjecture by Pavol Hell about circular coloring is solved in this article by showing that there is a function f ( ) for each : 0 < < 1 such that, if the odd-girth of a planar graph G is at least f ( ), then G is (2 + )-colorable. N
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)