Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2506.23237

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

Thomas Selig·Haoyue Zhu
View original source ↗Read source document ↗Export metadata ↓

Abstract

This work investigates the duality between two discrete dynamical processes: parking functions, and the Abelian sandpile model (ASM). Specifically, we are interested in the extension of classical parking functions, called $G$-parking functions, introduced by Postnikov and Shapiro in 2004. $G$-parking functions are in bijection with recurrent configurations of the ASM on $G$. In this work, we define a notion of prime $G$-parking functions. These are parking functions that are in a sense "indecomposable". Our notion extends the concept of primeness for classical parking functions, as well as the notion of prime $(p,q)$-parking functions introduced by Armon et al. in recent work. We show that from the ASM perspective, prime $G$-parking functions correspond to certain configurations of the ASM, which we call strongly recurrent. We study this new connection on a number of graph families, including wheel graphs, complete graphs, complete multi-partite graphs, and complete split graphs.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Thomas Selig, Haoyue Zhu. 2025-06-29. Prime graphical parking functions and strongly recurrent configurations of the Abelian sandpile model. https://arxiv.org/abs/2506.23237

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Adjunctions, Box Products, and Forcing Families

Sidorenko's conjecture states that the number of copies of any given bipartite graph in another graph of given density is asymptotically minimized by a random graph. For bipartite graphs containing a cycle, the forcing conjecture further asserts that asymptotic equality characterizes quasi-random graphs. We establish an adjoint identity for a general class of graph-substitution operators and use it to obtain Sidorenko and forcing results for balanced blow-ups, subdivisions, Cartesian products, and strong products.

math.CO↗
arXiv2026

On the Cost Number of Graphs with Determining Number Two

A distinguishing vertex coloring of a graph $G$ is a vertex coloring such that only the identity automorphism of $G$ preserves the coloring. A graph is $2$-distinguishable if it admits a distinguishing vertex coloring with two colors, and its cost $ρ(G)$ is the minimum size of a color class in such a coloring. The determining number of a graph $G$, denoted by $Det(G)$, is the minimum size of a subset $S\subseteq V(G)$ such that only the trivial automorphism fixes every element of $S$ pointwise. Boutin (J. Combin. Math. Combin. Comput. 85: 161-171, 2013) asked if $ρ(G)$ and $Det(G)$ can be arbitrarily far apart. While the case for $Det(G) = 1$ is trivial, the answer remained unknown for $Det(G) \ge 2$. In this manuscript, we show that if $Det(G)=2$ then not only is $ρ(G)$ bounded, but in fact $ρ(G) \leq 4$. This is the first resolution of Boutin's question for any nontrivial fixed determining number. Moreover, for every fixed $Det(G)= n$, we construct examples giving a lower bound on any possible upper bound for $ρ(G)$ in terms of $n$.

math.CO↗