This is due to Rayleigh scattering, i.e. the shorter wavelengths (those near the violet end of the visible spectrum) are more deflected by dust particles than the longer wavelengths (those near the red end of the visible spectrum).
When the sun is near the horizon, it the path of the light through the atmosphere (and in particular through layers which have a higher concentration of dust particles) is longer, thus the scattering away of the non-red components is more pronounced.
The sky does not skip over the green range of frequencies. The sky is green. Remove the scattered light from the Sun and the Moon and even the starlight, if you so wish, and you'll be left with something called airglow (check out the link, it's awesome, great pics, and nice explanation).
Because the link does such a good job explaining airglow, I'll skip the nitty gritty.
So you might be thinking, "Jim, you half-insane ceiling fan, everybody knows that the night sky is black!" Well, you're only half right. The night sky isn't black. The link above explains the science of it, but if that's not good enough, try to remember back to a time when you might have been out in the countryside. No bright city lights, just the night sky and trees. Now when you look at the horizon, can you see the trees? Yes, they're black silhouettes against the night sky. But how could you see black against black? The night sky isn't black. It's green thanks to airglow (or, if you're near a city, orange thanks to light pollution).
Stop, it's picture time. Here's an above the atmosphere view of the night sky from Wikipedia:
And one from the link I posted, just in case you didn't check it out:
See, don't be worried about green. The sky gets around to being green all the time.
Best Answer
There are two things in effect here: the sun's atmosphere and the earth's atmosphere.
The sun's appearance is determined by the structure of its atmosphere, which consists of the photosphere (average color temperature of around 5800K, i.e. what we perceive as pure neutral white), the chromosphere that is above it (which appears red) and the corona (which is extremely hot, > 1 million Kelvins, visible mainly in the X-ray).
The photosphere can be considered the surface of the sun (because you cannot see what is happening beneath it). It is what you see if you look directly at the sun's disk (with dark glasses) and it has the most intensity. The photosphere has a certain thickness, which means two things. First, as you can see in the diagram above, the temperature decreases significantly with height (about 2000 K), i.e. a little more red light will come from the outer parts of the photosphere, while a little more blue light will come from the inner parts of the photosphere. Second, if you look progressively near to the boundary of the sun's disk (more tangential to the sun), your line of sight will cut an ever longer distance out of the photosphere. Even though this line of sight section formally gets longer, the "penetration distance" of the photons, i.e. how "deep" we can see along this line section, stays about the same. Therefore, what we see will be progressively dominated by the upper photosphere and its lower temperatures if our sight approaches the sun's horizon. Hence, the sun will look a little redder close to its boundary. Note, that if the sun was considered an emitter that obeys Lambert's law, the sun's boundary would appear having about the same brightness as the center. However, since we see more of the cooler parts of the sun close to its boundary, and due to the fact that the total amount of black body radiation increases with temperature, we also see the boundary of the sun as being darker (in addition to it being redder). This effect is called Limb Darkening and the Wikipedia article repeats many arguments given here.
The chromosphere is more like a thin haze with much lower intensity, which is why you cannot see it if you look directly at the sun's disk. But you can see the red chromosphere if you look at the visible boundary of the sun, but of course you need at least an eclipse for that (or be in space), because otherwise our blue sky will mask it. This Wikipedia article specifically mentions solar eclipses as an opportunity to observe the chromosphere. Note, that while the photosphere is a thermal light source (with some absorption lines), the chromosphere is dominated by emission lines (mainly of hydrogen), so it cannot be reasonably described by a color temperature.
So, to summarize, the boundary of the visible sun appears a little redder than the center, because of the line of sight crossing a higher proportion of red emissions in the sun's atmosphere. This can be seen in the following picture.
But also the earth's atmosphere contributes to the appearance of the sun. At sunset, the sun's rays pass through a much higher distance through the atmosphere (because they are almost tangential to the earth's surface). This is somewhat similar to the abovementioned line of sight effect in the sun's atmosphere, at least geometrically. This causes much more Rayleigh scattering (you mentioned it) for these rays, i.e. blue light gets distracted, while red light passes more or less undisturbed to the viewer. Since the atmospheric distance travelled by the rays changes rapidly with the inclination angle, the earth's atmosphere acts like a color gradient filter which fades very steeply from almost white to red near the horizon. So if the sun approaches the horizon, at first only a little of its blue light gets filtered out, which makes its center appear yellowish and dimmer, so that the redder appearance near the boundary becomes more apparent to the eye. As the sun sets more and more, much more of the blue, green and some yellow light of the center gets attenuated, so that eventually not only the boundary appears red (which it has been all the time), but also the center.