## Abstract A proper coloring of the vertices of a graph is called a __star coloring__ if the union of every two color classes induces a star forest. The star chromatic number χ~__s__~(__G__) is the smallest number of colors required to obtain a star coloring of __G__. In this paper, we study the r
Girth of sparse graphs
✍ Scribed by Béla Bollobás; Endre Szemerédi
- Publisher
- John Wiley and Sons
- Year
- 2002
- Tongue
- English
- Weight
- 73 KB
- Volume
- 39
- Category
- Article
- ISSN
- 0364-9024
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✦ Synopsis
Abstract
For each fixed k ≥ 0, we give an upper bound for the girth of a graph of order n and size n + k. This bound is likely to be essentially best possible as n → ∞. © 2002 Wiley Periodicals, Inc. J Graph Theory 39: 194–200, 2002; DOI 10.1002/jgt.10023
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We prove that every graph of minimum degree at least r and girth at least 186 contains a subdivision of K rþ1 and that for r5435 a girth of at least 15 suffices. This implies that the conjecture of Haj ! o os that every graph of chromatic number at least r contains a subdivision of K r (which is fal