why do linoleun floors yellow with area rugs?
Chemical reaction between the rubber backing and the linoleum, some of the antioxidants used in the latex or rubber backing compound are of the 'staining' type (which are generally amine- or sulfur-containing) and could easily migrate to the residual unsaturation in your linoleum.
Two main things. One is that your area rugs may have solvents or fabric protectants (such as anti-stains or water repellants) that react with your linoleum. Another is that the linoleum that is not covered by the rugs have photo-oxidized (light fades it a little bit), leaving the ones under the rugs relatively newer or yellower than the exposed areas.