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从G-停车功能到B-停车功能

From G-parking functions to B-parking functions

Fengming Dong

备注：
31 pages, 10 figures, 2 tables and 30 references. https://www.sciencedirect.com/science/article/pii/S0097316518300815
发表信息：
Fengming Dong, From G-parking functions to B-parking functions. J. Combin. Theory Ser. A 160 (2018 Nov), 84-110
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摘要：

多重图$G=(V,E)$中的匹配$M$被称为是唯一受限的，如果$M$是由$V(M)$诱导的$G$的子图中的唯一完美匹配（即，由$M$饱和的顶点集）。对于$G$中的任何固定顶点$x_0$，存在从$G$的生成树集合到$S(G)-x_0$中大小为$|V|-1$的唯一限制匹配集合的双射，其中$S(G)$是通过细分$G$中的每条边而从$G$获得的二部图。因此，“二部图$H$的唯一限制匹配饱和部集$X$中的所有顶点”的概念可以被看作是“连通图中的生成树”的推广。受此启发，我们将连通多重图的“G-停车函数”的概念推广到具有二分图$(X,Y)$的二分图$H$的“B-停车函数”$f:X\rightarrow \{-1,0,1,2,\cdots \}$，并从$H$的唯一限制匹配集到$H$的B-停车函数集找到一个双射$ψ$.我们还表明，对于任何唯一限制的匹配$M$在$H$与$|M|=|X|$，如果$f=ψ(M)$，那么$\sum_{x\in X}f(x)$正是元素的数量$y\in Y-V(M)$这不是外部B-积极的$H$中的$M$，其中，新概念“$H$中关于$M$的外部B活动成员“是“关于连通多重图中生成树的外部活动边”。

英文摘要：

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 $ψ$ 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 $f=ψ(M)$, 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".

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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