Does touching women—whether they are mahrams or not—in public places cause sin if the woman who was touched is wearing perfume?

A woman is forbidden from wearing perfume when leaving her house if she passes by non-mahram (definition) men, based on the Prophet's (peace be upon him) saying: "Any woman who perfumes herself and passes by a group of people so that they may smell her fragrance is a an adulteress." As for touching someone who has applied perfume, there is nothing wrong with it. The prohibition applies specifically to the one who intended to apply the perfume, and touching usually does not transfer the scent. Even if it does, it is minimal and quickly dissipates. So, there is no need to worry about touching, and ignoring obsessive thoughts is an effective remedy for them.