A short cycle means a cycle of length at most 7. In this paper, we prove that planar graphs without adjacent short cycles are 3-colorable. This improves a result of Borodin et al. (2005).

Let G=(V,E,F) be a plane graph with the sets of vertices, edges, and faces V, E, and F, respectively. If one can color all elements in V∪E∪F using k colors so that any two adjacent or incident elements receive distinct colors, then G is said to be entirely k‐colorable. Kronk and Mitchem [Discrete Math 5 (1973) 253‐260] conjectured that every plane graph with maximum degree Δ is entirely (Δ+4)‐colorable. This conjecture has now been settled in Wang and Zhu (J Combin Theory Ser B 101 (2011) 490–501), where the authors asked: is every simple plane graph G≠K4 entirely (Δ+3)‐colorable? In this article, we prove that every simple plane graph with Δ≥8 is entirely (Δ+3)‐colorable, and conjecture that every simple plane graph, except the tetrahedron, is entirely (Δ+3)‐colorable.