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6 votes
1 answer
252 views

Pair matching between divisors less and more than $\sqrt{N}$

Let $n$ be the positive integer. Let $A$ and $B$ be sets of divisors of $n$ less and more than $\sqrt{n}$ respectively. Consider bipartite graph $(A, B)$, where two vertices are connected when one ...
3 votes
1 answer
377 views

Minimum planar bipartite graph to cover all perfect matching count

Given set $\mathcal T_n=\{0,1,\dots,2^n-1\}$ what is the minimum number of vertices $2m$ needed in a planar bipartite balanced graph such that at every $i\in\mathcal T_n$ there is a graph $G\in\...