In a moneyless market, a non storable, non transferable homogeneous commodity is reallocated between agents with single-peaked preferences. Agents are either suppliers or demanders. Transfers between a supplier and a demander are feasible only if they are linked, and the links form an arbitrary bipartite graph. Typically, supply is short in one segment of the market, while demand is short in another. Information about individual preferences is private, and so is information about feasible links: an agent may unilaterally close one of her links if it is in her interest to do so. Our egalitarian transfer solution rations only the long side in each market segment, equalizing the net transfers of rationed agents as much as permitted by the bilateral constraints. It elicits a truthful report of both preferences and links: removing a feasible link is never profitable to either one of its two agents. Together with efficiency, and a version of equal treatment of equals, these properties are characteristic.