Multiparking functions, graph searching, and the Tutte polynomial

Abstract

A parking function of length n is a sequence ( b 1 , b 2 , … , b n ) of non-negative integers for which there is a permutation π ∈ S n so that 0 ⩽ b π ( i ) < i for all i. A well-known result about parking functions is that the polynomial P n ( q ) , which enumerates the complements of parking functions by the sum of their terms, is the generating function for the number of connected graphs by the number of excess edges when evaluated at 1 + q . In this paper we extend this result to arbitrary connected graphs G. In general the polynomial that encodes information about subgraphs of G is its Tutte polynomial t G ( x , y ) , which is the generating function for spanning trees of G parametrized by external and internal activities of G. We define G-multiparking functions, which generalize the G-parking functions that Postnikov and Shapiro introduced in the study of certain quotients of the polynomial ring. We construct a family of algorithmic bijections between the spanning forests of a graph G and the G-multiparking functions. In particular, the bijection induced by the breadth-first search leads to a new characterization of external activity, and hence a representation of Tutte polynomial by the reversed sums of G-multiparking functions.