Prime graphical parking functions and strongly recurrent configurations of the Abelian sandpile model

Parking functions were originally introduced by Konheim and Weiss (1966) in their study of hashing functions. Since then, they have become a central object in combinatorics research, with a number of variations and generalisations. Prime parking functions are parking functions that cannot be decomposed into smaller parking functions. Building on recent work by Armon et al. (2024), we extend the notion of primeness to a generalisation known as graphical parking functions, or G-parking functions. Using the classical duality betwen G-parking functions and the Abelian sandpile model (ASM), we exhibit a bijection between prime G-parking functions and what we call strongly recurrent configurations of the ASM. We apply this to obtain various enumerative results for prime G-parking functions on graph families.