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# From G-parking functions to B-parking functions

Citation

Dong, F. (2018). From G-parking functions to B-parking functions. Journal of Combinatorial Theory, Series A, 160, 84-110. https://doi.org/10.1016/j.jcta.2018.06.007

Abstract

A matching

*M*in a multigraph*G=(V,E)*is said to be uniquely restricted if*M*is the only perfect matching in the subgraph of*G*induced by*V(M)*(i.e., the set of vertices saturated by*M*). For any fixed vertex χ_{0}in*G*, there is a bijection from the set of spanning trees of*G*to the set of uniquely restricted matchings of size |V|−1 in*S(G)*− χ_{0}, where*S(G)*is the bipartite graph obtained from*G*by subdividing each edge in*G*. Thus the notion “uniquely restricted matchings of a bipartite graph*H*saturating all vertices in a partite set*X*” can be viewed as an extension of “spanning trees in a connected graph”. Motivated by this observation, we extend the notion “G-parking functions” of a connected multigraph to “B-parking functions” ƒ:*X*→{−1,0,1,2,⋯} of a bipartite graph*H*with a bipartition (*X,Y*) and find a bijection ψ from the set of uniquely restricted matchings of*H*to the set of B-parking functions of*H*. We also show that for any uniquely restricted matching*M*in*H*with |M|=|X|, if ƒ=*ψ(M)*, then ∑_{x∈X}^{ƒ(x)}is exactly the number of elements*y∈Y−V(M)*which are not externally B-active with respect to*M*in*H*, where the new notion “externally B-active members with respect to*M*in*H”*is an extension of “externally active edges with respect to a spanning tree in a connected multigraph”.Date Issued

2018

Publisher

Elsevier

Journal

Journal of Combinatorial Theory, Series A

DOI

10.1016/j.jcta.2018.06.007

Description

This is the final draft, after peer-review, of a manuscript published in Journal of Combinatorial Theory, Series A. The published version is available online at https://doi.org/10.1016/j.jcta.2018.06.007