Ted,
If you can get it I would highly recommend doing the Vogelsang, Merced, Yos Val route. The last leg of this can be done in one day if desired; it's a long ways but essentially all downhill. Take the previously mentioned Yarts bus at 5:00PM back to Toulumne Meadows. Or, you can spend another night at LYV and really take your time the last day (it's a long time till 5:00PM). There are two routes from Vogelsang to Merced. I would recommend going over Vogelsang Pass and down the Lewis Creek trail. This affords an opportunity to do the small detour to Bernice Lake which, believe me, is well worth it.
I would also recommend calling the permit office right now to find out if by chance any advanced reservations are still available for the Rafferty Creek trail. Since it's 4th of July weekend there are probably no openings avaialble but you never know. It's worth a call. Others have recommended the Cathedral trail all the way to Merced. That is not only a very long haul for your first day, that trail is even more difficult to get a permit for than Rafferty Creek. Another very long first day trail option (shorter than Cathedral) is the Sunrise trail from the foot of Tenaya Lake. You take this one towards Clouds Rest and then down across the John Muir trail in a convoluted criss-cross route to Echo Valley before getting to Merced.
You mentioned camping sites. Vogelsang, Merced lake, and LYV all have designated camping areas specifically for backbackers which include composting toilets and bear boxes. The Merced Lake HSC also has showers you are entitled to use. The showers are over near the expensive High Sierra Camp tent cabins.
If, in the end you have no choice but to stand in line at the permit office then get there early because the line forms early. Some annoying people even camp out on the porch even though it's prohibited. I must really empahsize that since you are tackling this on a holiday weekend you may have to be very very flexible reagarding trail options and may likely end up going someplace entirely different than what you hoped for. That's the nature of holiday weekends. Do both yourself and everybody else waiting in line a favor by having a multitude of alternative hikes already prepared in your mind so you don't have to waste other people's time waiting in line behind you while you ponder different possibilities. If there is any way feasible to revise your trip plans I would highly recommend trying to hit the trails the middle of the week prior to the 4th or middle of the full week after the 4th. This will open many more possibilities for you. Again, try the advanced reservation phone line to find out is someting is available by reservation. In any event, we have a saying amoungst us. Wherever you end up hiking, it's all good!
Jim
Jim
Edited 3 time(s). Last edit at 03/25/2013 10:28AM by tomdisco.