Coloring graphs without fan vertex-minors and graphs without cycle pivot-minors

A fan Fk is a graph that consists of an induced path on k vertices and an additional vertex that is adjacent to all vertices of the path. We prove that for all positive integers q and k, every graph with sufficiently large chromatic number contains either a clique of size q or a vertex-minor isomorphic to Fk. We also prove that for all positive integers q and k≥3, every graph with sufficiently large chromatic number contains either a clique of size q or a pivot-minor isomorphic to a cycle of length k.