https://hal.inria.fr/hal-01207608Hopkins, SamSamHopkinsMIT - Massachusetts Institute of TechnologyPerkinson, DavidDavidPerkinsonReed CollegeBigraphical arrangementsHAL CCSD2014hyperplane arrangementsparking functionsabelian sandpile model[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM][MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO]Episciences Iam, CoordinationLouis J. Billera and Isabella Novik2015-10-01 09:29:122017-03-07 15:27:012015-10-01 09:32:39enConference papershttps://hal.inria.fr/hal-01207608/document10.46298/dmtcs.2398application/pdf1We 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. The full version of this paper is forthcoming in the $\textit{Transactions of the American Mathematical Society}$