S.B.
I have this in our rental property. It's caused by the rubber on the back of the mats, it reacts with the linoleum. Since it's a chemical reaction it is near impossible to get rid of...I tried EVERYTHING you can think of. The only thing that made any dent in it at all was scrubbing it with baking soda and lemon juice. You make a paste and then scrub it in to the floor with a stiff brush and then rinse with water. For me it didn't remove the stain completely, but it did lighten it considerably. You'll need new rugs that do not have a rubber back.