A special case of Hadwigerʹs conjecture

We investigate Hadwigerʹs conjecture for graphs with no stable set of size 3. Such a graph on at least 2 t − 1 vertices is not t − 1 colorable, so is conjectured to have a K t minor. There is a strengthening of Hadwigerʹs conjecture in this case, which states that there is a K t minor in which the preimage of each vertex of K t 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 K 1 9 n 4 / 5 minor using only vertices and single edges as preimages of vertices.