Algorithm for assigning people to groups
This problem is equivalent to Perfect Matching
We can view the input as an almost-complete graph, with L as its vertices and every two vertices connected by an edge except for those in C. We then want to find a set of edges that uses every vertex exactly once. This is the perfect matching problem.
To solve this problem, you can use any algorithm for finding a maximal matching, and check if the maximal matching has $n/2$ matches.