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    "description": "A graph G is 1-planar if it can be embedded in the plane in such a way \nthat each edge crosses at most one other edge. Borodin showed that \n1-planar graphs are 6-colorable, but his proof does not lead to a linear-time \nalgorithm. This paper presents a linear-time algorithm for 7-coloring \n1-plane graphs (which are 1-planar graphs already embedded in the plane). \nThe main difficulty in the design of our algorithm comes from \nthe fact that the class of 1-planar graphs is not closed under the \noperation of edge contraction. This difficulty is overcome by a structural \nlemma that may be useful in other problems on 1-planar graphs. This paper \nalso shows that it is NP-complete to decide whether a given 1-planar graph \nis 4-colorable. The complexity of the problem of deciding whether a given \n1-planar graph is 5-colorable is still unknown.", 
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