In Havel [2] (1969, Journal of Combinatorial Theory, 7:184~186), Havel provided that both 4-cycles and 5-cycles must be excluded to ensure that the plane graphs are 3-colorable.

• Havel给出两个反例；如果平面图含有4-圈或有5-圈是不可3-可着色的；因此4-圈和5-圈在证明平面图是3-可着色时必须排除.