Rubbing Alcohol
The first thing to try when removing permanent marker stains from your linoleum is rubbing alcohol, since you most likely have it on hand. To do this, place a few drops of the alcohol on a soft, clean white cloth and work it into the stain in a circular motion until it's gone. You can repeat the process several times to work the marker out. Wipe the area with soapy water and dry it thoroughly.
Mineral Spirits
Another product that may remove permanent marker stains is mineral spirits. You can try this method if the the rubbing alcohol didn't work. Place a few drops of mineral spirits onto a clean, soft white cloth and rub it into the stain in the same manner as the rubbing alcohol. The mineral spirits may be able to lift the mark out, especially if you get to it right away, when it's still fresh.
Lighter Fluid
If all else fails, give lighter fluid a try. Although it is a harsher chemical and has a stronger smell than the other two options, it may do the trick. Again, place a small amount of the fluid on a clean, soft white cloth and rub it into the marker stain. Repeat the process several times if need be, and wash the area with warm, soapy water to rinse the lighter fluid away. Dry the area thoroughly.
Linoleum Cleaning Dont's
In your quest to remove the permanent marker stain from the linoleum floor, avoid using products that may damage the flooring by possibly scratching or gouging it. These products include abrasive cleaners, such as scouring products, and scrubbing tools made with metal bristles.