Asymptotic giant branch (AGB) stars are known to produce "cosmic" fluorine, but it is uncertain whether these stars are the main producers of fluorine in the solar neighborhood or if any of the other proposed formation sites, Type II supernovae (SNe II) and/or Wolf-Rayet (W-R) stars, are more important. Recent articles have proposed both AGB stars and SNe II as the dominant sources of fluorine in the solar neighborhood. In this paper we set out to determine the fluorine abundance in a sample of 49 nearby, bright K giants for which we previously have determined the stellar parameters, as well as alpha abundances homogeneously from optical high-resolution spectra. The fluorine abundance is determined from a 2.3 mu m HF molecular line observed with the spectrometer Phoenix. We compare the fluorine abundances with those of alpha-elements mainly produced in SNe II and find that fluorine and the alpha-elements do not evolve in lockstep, ruling out SNe II as the dominating producers of fluorine in the solar neighborhood. Furthermore, we find a secondary behavior of fluorine with respect to oxygen, which is another evidence against the SNe II playing a large role in the production of fluorine in the solar neighborhood. This secondary behavior of fluorine will put new constraints on stellar models of the other two suggested production sites: AGB stars and W-R stars.