Bigraphical Arrangements

TL;DR

This paper introduces bigraphical arrangements, linking their regions' labels to parking functions, proving related conjectures, and providing bounds on the number of regions, thus advancing understanding of graph arrangements and their combinatorial properties.

Contribution

It defines bigraphical arrangements and proves conjectures connecting their regions' labels to parking functions, offering new insights and bounds in graph arrangement theory.

Findings

Proved conjectures relating bigraphical arrangements to parking functions.

Established a new proof of Stanley's bijection between labeled graphs and Shi arrangement regions.

Provided bounds on the number of regions in bigraphical arrangements.

Abstract

We define the bigraphical arrangement of a graph and show that the Pak-Stanley labels of its regions are the parking functions of a closely related graph, thus proving conjectures of Duval, Klivans, and Martin and of Hopkins and Perkinson. A consequence is a new proof of a bijection between labeled graphs and regions of the Shi arrangement first given by Stanley. We also give bounds on the number of regions of a bigraphical arrangement.