Planar graphs without adjacent cycles of prescribed lengths are 3-colorable∗

About Steinberg’s conjecture and Erd˝os’s open question on the 3coloring in planar graphs, it is left the following challenging question: whether every planar graph without cycles of length from 4 to 6 is 3-colorable. In order to ask the open problem, we prove that planar graphs without adjacent cycles of prescribed lengths are 3-colorable in which no i-cycle is adjacent to a j-cycle whenever 3 ≤ i ≤ j ≤ 6, every 5-cycle is adjacent to at most one 7-cycle and every 7-cycle is adjacent at most one 3-cycle. The result is an improvement of some results of Borodin et al (see [ Planar graphs without cycles of length from 4 to 7 are 3-colorable, J. Combin.Theory Ser. B 93 (2005) 303-311 ] and [ Planar graphs without adjacent cycles at most seven are 3-colorable, Discrete Math. 310 (2010) 167-173 ]). MSC: 05C15, 05C69