  We study the subleading-color (nonplanar) contributions to the four-gluon
scattering amplitudes in N=4 supersymmetric SU(N) Yang-Mills theory. Using the
formalisms of Catani and of Sterman and Tejeda-Yeomans, we develop explicit
expressions for the infrared-divergent contributions of all the
subleading-color L-loop amplitudes up to three loops, and make some conjectures
for the IR behavior for arbitrary L. We also derive several intriguing
relations between the subleading-color one- and two-loop four-gluon amplitudes
and the four-graviton amplitudes of N=8 supergravity. The exact one- and
two-loop N=8 supergravity amplitudes can be expressed in terms of the one- and
two-loop N-independent N=4 SYM amplitudes respectively, but the natural
generalization to higher loops fails, despite having a simple interpretation in
terms of the 't Hooft picture. We also find that, at least through two loops,
the subleading-color amplitudes of N=4 SYM theory have uniform
transcendentality (as do the leading-color amplitudes). Moreover, the N=4 SYM
Catani operators, which express the IR-divergent contributions of loop
amplitudes in terms of lower-loop amplitudes, are also shown to have uniform
transcendentality, and to be the maximum transcendentality piece of the QCD
Catani operators.
