An exchange network is a socioeconomic system in which any two actors are allowed to negotiate and conclude a transaction if and only if their positions-mathematically represented by the points of a connected graph-are joined by a line of this graph. A transaction consists in a bilaterally agreed-on division of a profit pool assigned to a given line. Under the one-exchange rule, every actor is permitted to make no more than one transaction in each negotiation round. Bienenstock and Bonacich () proposed to represent a one-exchange network with an n-person game in characteristic function form. A special case, known as a two-sided assignment game, was studied earlier by Shapley and Shubik () who proved that the game representing any one-exchange network has a nonempty core if the underlying graph is bipartite. This paper offers a graph-theoretic criterion for the existence of a nonempty core in the game associated with an arbitrary not necessarily bipartite homogeneous one-exchange network where network homogeneity means that every line of the transaction opportunity graph is assigned a profit pool of the same size.