A special case of Hadwiger's conjecture

We investigate Hadwiger's conjecture for graphs with no stable set of size 3. Such a graph on at least 2t−1 vertices is not t−1 colorable, so is conjectured to have a Kt minor. There is a strengthening of Hadwiger's conjecture in this case, which states that there is a Kt minor in which the preimage of each vertex of Kt is a single vertex or an edge. We prove this strengthened version for graphs with an even number of vertices and fractional clique covering number less than 3. We investigate several possible generalizations and obtain counterexamples for some and improved results from others. We also show that for sufficiently large n, a graph on n vertices with no stable set of size 3 has a minor using only vertices and single edges as preimages of vertices.