Abstract: A graph G is called 1-planar graph if it can be drawn on the plane so that each edge is crossed by at most one other edge. A propervertexcoloring of G is an assignment φ of integer to the vertex of G such that φ(u)≠φ(v)if the vertex u and v adjacent in G. A k-coloring is a proper vertex coloring using k colors at least. The theorem that every 1-planar graph without cycles of length 3 or 4 is 5-colorable was given by the Discharging Method.

Key words: cross vertices, cross faces, k-colorable, 1-planar graph