In the Gossiping problem, the edges of a multigraph $G$ are calls between participants, and one asks for an ordering of the calls, given by a labeling of the edges with natural numbers, such that every participant learns the secret of every other participant. Equivalently, the edge-labeled graph must be temporally connected: for every ordered pair of vertices there is a path whose edge labels strictly increase. We say that $G$ is label-connected if such a labeling exists.

This submission defines the problem and the two structural witnesses of the classical characterization of label-connected multigraphs. A $T$-graph has two spanning trees sharing at most one edge. An $M$-graph is not a $T$-graph but contains a cycle of length at least four together with two edge-disjoint spanning forests, each with exactly four components, all of which meet the cycle. The main statement is that a finite multigraph is label-connected if and only if it has at most one vertex, is a $T$-graph, or is an $M$-graph.