Name for a complete graph with a perfect matching removed?
Solution 1:
Generally speaking, it's a special case of a Turán graph, because it's the complete $n$-partite graph $K_{2,2,2,\dots,2,2}$.
This particular Turán graph is also known as a cocktail party graph, representing the number of handshakes that occur if $n$ couples go to a cocktail party and everyone shakes hands with everyone they meet there (but not with their partner).