Totally Odd-subdivisions in 4-chromatic Graphs
β Scribed by Carsten Thomassen
- Publisher
- Springer-Verlag
- Year
- 2001
- Tongue
- English
- Weight
- 363 KB
- Volume
- 21
- Category
- Article
- ISSN
- 0209-9683
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π SIMILAR VOLUMES
It is shown that any 4-chromatic graph on n vertices contains an odd cycle of length smaller than β 8n.
## Abstract It is shown that every 4βchromatic graph on __n__ vertices contains an odd cycle of length less than $2\sqrt {n}\,+3$. This improves the previous bound given by Nilli [J Graph Theory 3 (1999), 145β147]. Β© 2001 John Wiley & Sons, Inc. J Graph Theory 37: 115β117, 2001
It is known that the Mycielski graph can be generalized to obtain an infinite family of 4-chromatic graphs with no short odd cycles. The first proof of this result, due to Stiebitz, applied the topological method of Lov~sz. The proof presented here is elementary combinatorial.
The total chromatic number XT(G) of a graph G is the least number of colours needed to colour the edges and vertices of G so that no incident or adjacent elements receive the same colour. This paper shows that if G is odd order and regular of degree d > [(&? -1)/6]1 V(G)/, then a necessary and suffi