Induced subgraphs of power and commuting graphs

For those who like thinking about these things, here is a small observation and a few problems.

As I have recently discussed, the power graph of a group is perfect. This means that all its induced subgraphs are perfect, and in particular they all have clique number equal to chromatic number.

Problem 1 Is every perfect graph an induced subgraph of the power graph of some group?

For the commuting graph, things are different: every finite graph is an induced subgraph of the commuting graph of a group. To see this we use the following simple construction. Let V be a vector space over the field F with two elements. Let B be a bilinear form on V. Define an operation on V×F by the rule

(v,a) * (w,b) = (v+w,a+b+B(v,w).

It is easily checked that this is a group operation. Now let {v₁,…v_n} be a basis for V. The values of B(v_i,v_j) can be prescribed arbitrarily, and the resulting function extended to V×V by bilinearity. So, given a graph on the vertex set {1,…n}, put B(v_i,v_j) equal to 0 if i ≤ j, while for i > j put the value equal to 0 if i and j are joined, 1 otherwise. This works because the commutator of (v,0) and (w,0) is (0,B(v,w)+B(w,v)).

Problem 2 Given a graph on n vertices, consider the group defined above. Which other graphs can be realised by changing the basis for V?

It follows that, for every n, there is a group whose commuting graph contains every n-vertex graph as an induced subgraph.

Problem 3 What is the smallest group G with this property?

Problem 4 Is every graph an induced subgraph of the enhanced power graph of a group?

Problem 5 Is every graph an induced subgraph of the generating graph of a 2-generator group?