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Totally Odd-subdivisions in 4-chromatic Graphs

✍ 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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Short odd cycles in 4-chromatic graphs
✍ Nilli, A. πŸ“‚ Article πŸ“… 1999 πŸ› John Wiley and Sons 🌐 English βš– 163 KB πŸ‘ 3 views

It is shown that any 4-chromatic graph on n vertices contains an odd cycle of length smaller than √ 8n.

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## Abstract It is shown that every 4‐chromatic 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), 145–147]. Β© 2001 John Wiley & Sons, Inc. J Graph Theory 37: 115–117, 2001

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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.

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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