English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We study the minimum number of maximum matchings in a bipartite multigraph Gwith parts $X$ and $Y$ under various conditions, refining the well-known lowerbound 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 minimumis $r!(k-r+1)$ 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 $|Y|-|X|=t\ge 0$, theminimum is $[(n-1)t+2+b](t+1)$, where $b=|E(G)|-2(n+t)$. We also determine theminimum number of maximum matchings in several other situations. We provide avariety of sharpness constructions.

Sharp lower bounds for the number of maximum matchings in bipartite multigraphs