On metric graphs with prescribed gonality

We prove that in the moduli space of genus-g metric graphs the locus of graphs with gonality at most d has the classical dimension min⁡{3g−3,2g+2d−5}. This follows from a careful parameter count to establish the upper bound and a construction of sufficiently many graphs with gonality at most d to establish the lower bound. Here, gonality is the minimal degree of a non-degenerate harmonic map to a tree that satisfies the Riemann–Hurwitz condition everywhere. Along the way, we establish a convenient combinatorial datum capturing such harmonic maps to trees.