A matching of a graph is a set of its edges which do not share any vertices. This can be further qualified as e.g. a maximal matching, perfect matching, etc.
I was wondering if the converse statement is a well-studied notion. That is, I am interested in the set of edges which all pairwise share a vertex.
For instance, consider the complete graph $K_4$ (the context of my interest is in complete, undirected graphs, though such a restriction is not necessary for this question). The subsets $E_1 =\{12,13,14\}$ and $E_2 =\{23,24,34\}$ are valid partitionings of the edges such that, within each, a vertex is pairwise shared (not necessarily the same vertex for the entire set, as seen in $E_2$), and they satisfy $$E_1 \cap E_2 = \varnothing,$$ $$E_1 \cup E_2 = E.$$ Is there a common terminology for this idea, so that I know what phrase to search for? I have read about related notions, such as cliques and covers, but they are not quite what I have in mind.
(For what it's worth, I conjecture that the smallest number of such partitionings for $K_n$ is $n-2$, or at least $O(n)$, but my construction is quite naive and I have not been able to prove optimality. I suspect that this would be a classically studied problem, but I have not seen anything so far in my elementary graph theory text or online.)