Multiplicity-free gonality on graphs

Frances Dean (Department of Mathematics, University of California Berkeley, Berkeley, CA, USA)
Max Everett (Department of Mathematics, City University of New York, New York, NY, USA)
Ralph Morrison (Departments of Mathematics and Statistics, Williams College, Williamstown, MA 01267)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 11, No 2 (2023): Electronic Journal of Graph Theory and Applications

The divisorial gonality of a graph is the minimum degree of a positive rank divisor on that graph. We introduce the multiplicity-free gonality of a graph, which restricts our consideration to divisors that place at most 1 chip on each vertex. We give a sufficient condition in terms of vertex-connectivity for these two versions of gonality to be equal; and we show that no function of gonality can bound multiplicity-free gonality, even for simple graphs. We also prove that multiplicity-free gonality is NP-hard to compute, while still determining it for graph families for which gonality is currently unknown. We also present new gonalities, such as for the wheel graphs.