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 < 0.99log 2n, 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 ≤ 0.99log 3p, 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/logp) and for k = Θ (p) it is Θ(plogp).For some non-abelian groups (like SL2 (Z q) ), the chromatic number is, asymptotically almost surely, at least kΩ (1) for every k, whereas for elementary abelian 2-groups of order n = 2t and any k satisfying 1.001t ≤ k ≤ 2.999t 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.
All Science Journal Classification (ASJC) codes
- Discrete Mathematics and Combinatorics