Parking Functions on Directed Graphs and Some Directed Trees

Classical parking functions can be defined in terms of drivers with preferred parking spaces searching a linear parking lot for an open parking spot. We may consider this linear parking lot as a collection of $n$ vertices (parking spots) arranged in a directed path. We generalize this notion to allow for more complicated "parking lots" and define parking functions on arbitrary directed graphs. We then consider a relationship proved by Lackner and Panholzer between parking functions on trees and "mapping digraphs" and we show that a similar relationship holds when edge orientations are reversed.