From G-parking functions to B-parking functions

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 $x_0$ 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 $S(G)-x_0$, 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" $f:X\rightarrow \{-1,0,1,2,\cdots \}$ of a bipartite graph $H$ with a bipartition $(X,Y)$ and find a bijection $\psi$ 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 $f=\psi(M)$, then $\sum_{x\in X}f(x)$ is exactly the number of elements $y\in Y-V(M)$ 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".