(L-R) David Zaslav and Graydon Carter attend the Cannes Film Festival Air Mail Party at Hotel du Cap-Eden-Roc on May 23, 2023 in Cap d'Antibes, France. Getty Images for Air Mail/Warner Brothers Discovery Forget the movies! The biggest buzz at the Cannes Film Festival so far centers on dueling parties between  Graydon Carter  and his former magazine, Vanity Fair. Like its most recent Oscars party, Vanity Fair is taking a downsized approach this year in the south of France, Page Six Hollywood has learned. Carter, meanwhile, is holding his affair at the superluxe Hôtel du Cap with the backing of a Silicon Valley billionaire and Hollywood’s biggest connector.    Ex-Vanity Fair boss Graydon Carter is throwing his own party. WireImage Said a source, “VF is doing a smaller dinner with a cocktail following,” without elaborating on the number of guests.…