On 3-colorable planar graphs without short cycles

In this paper, we prove that planar graphs without {4,6,7}{4,6,7}-cycles and without two 5-cycles sharing exactly one edge are 3-colorable. This improves the result in [O.V. Borodin, A.N. Glebov, A. Raspaud, M.R. Salavatipour, Planar graphs without cycles of length from 4 to 7 are 3-colorable, J. Combin. Theory Ser. B 93 (2005) 303–311].