A graph G is 1-planar if it can be embedded in the plane in such a way that each edge crosses at most one other edge. Borodin showed that 1-planar graphs are 6-colorable, but his proof only leads to a complicated polynomial (but nonlinear) time algorithm. This paper presents a linear-time algorithm for 7-coloring 1-planar graphs (that are already embedded in the plane). The main difficulty in the design of our algorithm comes from the fact that the class of 1-planar graphs is not closed under the operation of edge contraction. This difficulty is overcome by a structure lemma that may find useful in other problems on 1-planar graphs. This paper also shows that it is NP-complete to decide whether a given 1-planar graph is 4-colorable. The complexity of the problem of deciding whether a given 1-planar graph is 5-colorable is still unknown.
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