Disclaimer: Yes, there have been claims of Presidents having Mafia ties. All have been unsubstantiated, and/or are just popular myth said a dozen times over, with all the times they've been said used to say they are well known fact and without evidence to back them up. One of the most well known are the Kennedy's, which is a myth: Joe Sr was not a bootlegger nor involved with mafia figures as a result. I also know of a claim about Nixon. For the purposes of this thread, none of these claims will be treated as any more than myth and urban legend as they most likely are anyway.
The challenge here is to get a United States politician to the highest office in the land who has ties to Organized Crime.