Small genus-4 Lefschetz fibrations on simply-connected 4-manifolds

We consider simply connected 4-manifolds admitting Lefschetz fibrations over the 2-sphere. We explicitly construct nonhyperelliptic and hyperelliptic Lefschetz fibrations of genus 4 on simply-connected 4-manifolds which are exotic symplectic 4-manifolds in the homeomorphism classes of CP2 #8 (CP2) over bar and CP2#9 (CP2) over bar, respectively. From these, we provide upper bounds for the minimal number of singular fibers of such fibrations. In addition, we prove that this number is equal to 18 for g = 3 when such fibrations are hyperelliptic. Moreover, we discuss these numbers for higher genera.