Ive read painter and decorators use sugar soap to get rid of heavy smoke film/damage from paintwork so I should imagine it would remove it from your floors. Do first carry out a test sample in a discrete corner of the floor as I am not sure what it would be like on tiles. All the best.
The floors are yellowed because of wax build up. Use one cup of vinegar and one cup of ammonia to one gallon of warm water. mop this mixture over the floor and allow it to sit for a few mins to loosen the wax, then use a scrub brush to finish removing the wax. Mop up and use clean water to rinse. It sounds harder than it is. Don't use comet or any other abrasive cleanser on linoleum floors. And the last thing I want to mention is that the linoleum could be worn out and in need of replacement. Hope this helps.
could it be built up wax? when nicotine gets into things it's almost imposable to get rid of ALL of it. i like ORANGE FANTASTIC and a scrub brush. for wax you'll need some kind of a stripper and buffer. good luck on with that problem.
Now THAT would be a LOT of cigarettes. Are you sure it's not just old? Or maybe someone spilled something on it?
On tile i use bleach, linoleum i use mean green both work very well.