Israeli citizen Rotem Gez, spray-painted bloody handprints and the slogan "blood on your hands" on the social media giant's building at the Rothschild Boulevard street, the Jerusalem Post reported on Sunday.
The man later told a local TV channel that he vandalised the building because Facebook "doesn't take down pages that call for the murder of Israelis", the daily said.
He also alleged that the company keeps controversial anti-semitic pages online because they were good for traffic.
Police said no complaint about the graffiti was received and that they were not familiar with the case.
In 2011, Gez legally changed his name to "Mark Zuckerberg", so that in a lawsuit Facebook was filing against him they would have to say they were suing someone with the same name as their founder.
The company sued Gez over a system he created to sell "likes" for posts that users make on the site.
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