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 nonnegative 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 the Tutte polynomial t G (x, y), which is the generating function for two parameters, namely the internal and external activities, associated with the spanning trees 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 sum of G-multiparking functions.

Combinatorial interpretations for TG(1, −1)

“…In particular, for any spanning tree T of G , let π T be a sequence on vertices of G formed by the vertex‐adding order, which is the third example of a proper tree order given in Chebikin and Pylyavskyy , and described in Algorithm A below. With respect to the sequence π T , we define N T as the set of T ‐redundant edges of G , in the sense of Kostić and Yan , Section 4.1 and as defined below after Algorithm A. In , p 91 it is shown that .…”

“…With respect to the sequence π T , we define N T as the set of T ‐redundant edges of G , in the sense of Kostić and Yan , Section 4.1 and as defined below after Algorithm A. In , p 91 it is shown that . We then proceed to define even spanning trees and left spanning trees.…”

A Characterization of the Tutte Polynomial via Combinatorial Embeddings

“…For instance, a notion of external DFS-activity inspired by the depth-first search algorithm was investigated in [12][13][14] and a notion of external BFS-activity inspired by the breadth-first search algorithm was investigated in [16]. The definitions of external BFSand DFS-activities require a choice of a linear order on the vertex set.…”

Bijections Between Multiparking Functions, Dirichlet Configurations, and Descending R-Traversals

“…Kostić and Yan [11] generalized this work further. A G-multiparking function is a function f : V (G) → N ∪ {∞} such that for any U ⊆ V (G) there exists i ∈ U with either (A) f (i) = ∞, or (B) 0 ≤ f (i) < O U (i).…”

“…There is a subtle but important difference between this definition of G-multiparking function and the one that appears in [11]. In that paper, the minimal vertex in each component of G is required to be a root; here there is no such restriction.…”