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On friendship and cyclic parking functions

Yujia Kang, Thomas Selig, Guanyi Yang, Yanting Zhang, Haoyue Zhu

TL;DR

This work introduces a variant of classical parking functions, called"friendship parking functions", which imposes additional restrictions on where cars can park, and characterise and enumerate such friendship parking functions according to their outcome permutation, which describes the final configuration when all cars have parked.

Abstract

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

On friendship and cyclic parking functions

TL;DR

This work introduces a variant of classical parking functions, called"friendship parking functions", which imposes additional restrictions on where cars can park, and characterise and enumerate such friendship parking functions according to their outcome permutation, which describes the final configuration when all cars have parked.

Abstract

In parking problems, a given number of cars enter a one-way street sequentially, and try to park according to a specified preferred spot in the street. Various models are possible depending on the chosen rule for collisions, when two cars have the same preferred spot. In classical parking functions, if a car's preferred spot is already occupied by a previous car, it drives forward and looks for the first unoccupied spot to park. In this work, we introduce a variant of classical parking functions, called "friendship parking functions", which imposes additional restrictions on where cars can park. Namely, a car can only end up parking next to cars which are its friends (friendship will correspond to adjacency in an underlying graph). We characterise and enumerate such friendship parking functions according to their outcome permutation, which describes the final configuration when all cars have parked. We apply this to the case where the underlying friendship graph is the cycle graph. Finally, we consider a subset of classical parking functions, called "cyclic parking functions", where cars end up in an increasing cyclic order. We enumerate these cyclic parking functions and exhibit a bijection to permutation components.
Paper Structure (10 sections, 14 theorems, 2 equations, 5 figures, 1 table)

This paper contains 10 sections, 14 theorems, 2 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Parking functions and their variations
  3. Friendship parking functions
  4. Contents of the paper
  5. Characterising and counting friendship parking functions
  6. Friendship parking functions and Hamiltonian paths
  7. Characterising friendship parking functions according to their outcomes
  8. Application: the cycle graph case
  9. Cyclic parking functions and permutation components
  10. Conclusion and future work

Key Result

Proposition 2.1

Let $G=(V,E)$ be a graph. Then the set of $G$-friendship parking functions is non-empty if, and only if, $G$ contains a Hamiltonian path.

Figures (5)

  • Figure 1: Illustrating the parking process for the classical parking function $p = (3,1,1,2)$.
  • Figure 2: Illustrating the parking process for the friendship parking function $p = (2,1,2,2)$.
  • Figure 3: Illustrating the friendship parking process for the parking preference $p = (4,2,2,1)$. Here, car $3$ is unable to park, so $p$ is not a friendship parking function.
  • Figure 4: The graph $G$ with Hamiltonian path $\pi=87152463$ (red)
  • Figure 5: The cycle graph $C_6$ on six vertices.

Theorems & Definitions (40)

  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.4
  • ...and 30 more