  We consider the problem of distributed lossless computation of a function of
two sources by one common user. To do so, we first build a bipartite graph,
where two disjoint parts denote the individual source outcomes. We then project
the bipartite graph onto each source to obtain an edge-weighted characteristic
graph (EWCG), where edge weights capture the function's structure, by how much
the source outcomes are to be distinguished, generalizing the classical notion
of characteristic graphs. Via exploiting the notions of characteristic graphs,
the fractional coloring of such graphs, and edge weights, the sources
separately build multi-fold graphs that capture vector-valued source sequences,
determine vertex colorings for such graphs, encode these colorings, and send
them to the user that performs minimum-entropy decoding on its received
information to recover the desired function in an asymptotically lossless
manner. For the proposed EWCG compression setup, we characterize the
fundamental limits of distributed compression, verify the communication
complexity through an example, contrast it with traditional coloring schemes,
and demonstrate that we can attain compression gains higher than $\% 30$ over
traditional coloring.
