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 disks 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.
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