Planar graphs without 5- and 7-cycles and without adjacent triangles are 3-colorable (Q1026007)

\textit{0. V. Borodin}, \textit{A. N. Glebvov}, \textit{A. Raspaud} and \textit{M. R. Salavatipour} [J. Comb. Theory, Ser. B 93, No. 2, 303--311 (2005; Zbl 1056.05052)] showed that planar graphs without cycles of length 4, 5, 6, or 7 are 3-colorable. The present authors improve this result by proving that planar graphs without cycles of length 5 or 7 and also without adjacent triangles are 3-colorable. They also give counterexamples to the argument given for the same result by \textit{B. Xu} [J. Comb. Theory, Ser. B 96, No. 6, 958--963 (2006; Zbl 1108.05046)].