  We present Galaxy Evolution Explorer (GALEX) far-UV and near-UV mosaic
observations covering the entirety of M31 and M33. For both targets, we measure
the decline of surface brightness (in FUV and NUV) and changes in FUV--NUV
color as a function of galactocentric radius. These UV radial profiles are
compared to the distribution of ionized gas traced by H-alpha emission. We find
that the extent of the UV emission, in both targets, is greater than the extent
of the observed HII regions and diffuse ionized gas. We determine the
ultraviolet diffuse fraction in M33 using our FUV observations and compare it
to the H-alpha diffuse fraction obtained from wide-field narrow-band imaging.
The FUV diffuse fraction appears to be remarkably constant near 0.65 over a
large range in galactocentric radius, with departures to higher values in
circumnuclear regions and, most notably, at the limit of the H-alpha disk. We
suggest that the increase in FUV diffuse fraction at large galactocentric radii
could indicate that a substantial portion of the diffuse emission beyond this
point is not generated in situ but rather scattered from dust, after
originating in the vicinity of the disk's outermost HII regions. Radial
variation of the H-alpha diffuse fraction was also measured. We found the
H-alpha diffuse fraction generally near 0.4 but rising toward the galaxy
center, up to 0.6. We made no attempt to correct our diffuse fraction
measurements for position-dependent extinction, so the quoted values are best
interpreted as upper limits given the plausibly higher extinction for stellar
clusters relative to their surroundings.
