The chromatic number of random Cayley graphs

We consider the typical behavior of the chromatic number of a random Cayley graph of a given group of order n with respect to a randomly chosen set of size k ≤ n / 2 . This behavior depends on the group: for some groups it is typically 2 for all k < 0.99 log 2 n , whereas for some other groups it grows whenever k grows. The results obtained include a proof that for any large prime p , and any 1 ≤ k ≤ 0.99 log 3 p , the chromatic number of the Cayley graph of Z p with respect to a uniform random set of k generators is, asymptotically almost surely, precisely 3. This strengthens a recent result of Czerwiński. On the other hand, for k > log p , the chromatic number is typically at least Ω ( k / log p ) and for k = Θ ( p ) it is Θ ( p log p ) . me non-abelian groups (like S L 2 ( Z q ) ), the chromatic number is, asymptotically almost surely, at least k Ω ( 1 ) for every k , whereas for elementary abelian 2-groups of order n = 2 t and any k satisfying 1.001 t ≤ k ≤ 2.999 t the chromatic number is, asymptotically almost surely, precisely 4 . Despite these discrepancies between different groups, it seems plausible to conjecture that for any group of order n and any k ≤ n / 2 , the typical chromatic number of the corresponding Cayley graph cannot differ from k by more than a poly-logarithmic factor in n .