On October 11, the U.S. Supreme Court refused to hear Libertarian Party of New Hampshire v Gardner, 11-119. This is the case in which the First Circuit had said the U.S. Constitution does not provide any protection for party names on general election ballots, and if a state puts a candidate on the ballot with a party label, and the party had not nominated that candidate (but had nominated another candidate for the same office), the party has no recourse.
In November 2008, New Hampshire listed two Libertarians on the ballot for President, one whom the state and national party had nominated (Bob Barr) and someone else. The 10th circuit had ruled in 1984 that states must give parties protection for their names. There aren’t many lawsuits on this subject, because almost all states do provide party name protection.
The Court thus continues its 20-year behavior of refusing to hear any case brought by a minor party or independent candidate, unless a major party is also in the same case on the same side as the minor party or independent candidate (excluding a 1996 case brought by the Georgia Libertarian Party on whether states can force all candidates for state office to be tested for illegal drugs, which was really a Fourth Amendment case, not an election law case). During those same 20 years, however, the Court accepted three cases in which the minor party or independent candidate had won the case below, and reversed them. They had been brought by the states of Minnesota, Arkansas, and Oklahoma, and involved fusion, entrance into candidate debates, and whether a party may insist on an open primary for itself.