Multiplicity-Free Gonality on Graphs

"In this thesis we study a new graph invariant which we call multiplicity-free gonality, a relative of divisorial gonality where we are restricted to consideration of divisors that map the vertices of a graph G to {0,1}. We present a condition to guarantee equality between multiplicity-free gonality and divisorial gonality, as well as a proof demonstrating that divisorial gonality cannot bound multiplicity-free gonality for simple graphs. We use Dhar's Burning Algorithm to prove both of these results. We also present families of graphs with previously-known gonality and multiplicity-free gonality to demonstrate the similarities and differences between these invariants, but we present a novel proof using scramble number, a new graph invariant that acts as a lower bound to gonality."

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