论文标题
尖锐的下边界，用于两部分多编码中最大匹配的数量
Sharp lower bounds for the number of maximum matchings in bipartite multigraphs
论文作者
论文摘要
我们在各种条件下研究了两分之一的$ x $和$ y $的两部分的多式G中的最小匹配数，并促进了由于M. Hall而闻名的下限。当$ | x | = n $时，$ x $中的每个顶点至少具有至少$ k $，并且$ x $中的每个顶点至少具有至少$ r $不同的邻居，当$ r！（k-r+1）$当$ n \ ge r $和$ n \ ge r $和is $ [r+n（k-r+n（k-k）]当每个顶点至少有两个邻居和$ | y | - | x | = t \ ge 0 $，最小值为$ [（n-1）t+2+b]（t+1）$，其中$ b = | e（g）| -2（n+t）$。我们还确定了其他几种情况下最大匹配的最小匹配数量。我们提供各种清晰度的结构。
We study the minimum number of maximum matchings in a bipartite multigraph G with parts $X$ and $Y$ under various conditions, refining the well-known lower bound due to M. Hall. When $|X|=n$, every vertex in $X$ has degree at least $k$, and every vertex in $X$ has at least $r$ distinct neighbors, the minimum is $r!(k-r+1)$ when $n\ge r$ and is $[r+n(k-r)]\prod_{i=1}^{n-1}(r-i)$ when $n<r$. When every vertex has at least two neighbors and $|Y|-|X|=t\ge 0$, the minimum is $[(n-1)t+2+b](t+1)$, where $b=|E(G)|-2(n+t)$. We also determine the minimum number of maximum matchings in several other situations. We provide a variety of sharpness constructions.
