Given an undirected graph $G=(V,E)$, use mathematical induction to show how to construct a graph with $2n$ nodes and $n^2$ edges such that the graph has exactly one, unique, complete pairing.
Note: A pairing is a set $P⊆E$ of edges such that for all $(u,v),(x,y)∈P$, the nodes $u, v, x, y$ are all different. In other words, no two edges in $P$ have a node in common. A complete pairing is a pairing P that uses all the graph’s nodes, that is, a pairing for which
I'm struggling a lot with how to approach this problem. I do understand what a pairing is and I can think of some example graphs with $2n$ nodes and $n^2$ edges s.t. there is a complete pairing (I've drawn a few) but I'm unsure how to approach it in terms of mathematical induction. I can't seem to figure out a base case without drawing a graph or how to approach it mathematically. Any ideas? I would really appreciate any help, hints or a start.