On friendship and cyclic parking functions

In parking problems, a given number of cars enter a one-way streetsequentially, and try to park according to a specified preferred spot in thestreet. Various models are possible depending on the chosen rule forcollisions, when two cars have the same preferred spot. In classical parkingfunctions, if a car's preferred spot is already occupied by a previous car, itdrives forward and looks for the first unoccupied spot to park. In this work,we introduce a variant of classical parking functions, called "friendshipparking functions", which imposes additional restrictions on where cars canpark. Namely, a car can only end up parking next to cars which are its friends(friendship will correspond to adjacency in an underlying graph). Wecharacterise and enumerate such friendship parking functions according to theiroutcome permutation, which describes the final configuration when all cars haveparked. We apply this to the case where the underlying friendship graph is thecycle graph. Finally, we consider a subset of classical parking functions,called "cyclic parking functions", where cars end up in an increasing cyclicorder. We enumerate these cyclic parking functions and exhibit a bijection topermutation components.