# Vertex transitive graphs

Does having vertex transitivity make the problem of calculating independence and chromatic numbers easier?

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Not particularly. There is a paper by Codenotti, Gerace, Vigna "Hardness results and spectral techniques for combinatorial problems on circulant graphs" Linear Algebra Appl. 285 (1998) 123-142 which shows that computing the chromatic number of a circulant graph is NP-hard. (The pdf is available on Codenotti's web page.) Being vertex transitive guarantees that a $k$-regular graph has vertex connectivity at at least $2(k+1)/3$ and that its edge connectivity is equal to $k$. Aside from this, it is not easy to identify useful consequences of vertex transitivity.
Certainly, in a particular case transitivity can make a big difference. But I was thinking of theorems running "if $G$ is vertex transitive, then..." – Chris Godsil Apr 2 '12 at 19:35