  We construct a diagrammatic coaction acting on one-loop Feynman graphs and
their cuts. The graphs are naturally identified with the corresponding (cut)
Feynman integrals in dimensional regularization, whose coefficients of the
Laurent expansion in the dimensional regulator are multiple polylogarithms
(MPLs). Our main result is the conjecture that this diagrammatic coaction
reproduces the combinatorics of the coaction on MPLs order by order in the
Laurent expansion. We show that our conjecture holds in a broad range of
nontrivial one-loop integrals. We then explore its consequences for the study
of discontinuities of Feynman integrals, and the differential equations that
they satisfy. In particular, using the diagrammatic coaction along with
information from cuts, we explicitly derive differential equations for any
one-loop Feynman integral. We also explain how to construct the symbol of any
one-loop Feynman integral recursively. Finally, we show that our diagrammatic
coaction follows, in the special case of one-loop integrals, from a more
general coaction proposed recently, which is constructed by pairing master
integrands with corresponding master contours.
