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，希伍德猜想提供的上限也给出了必要的颜色数，但克莱因瓶（希伍德公式给出 7，但正确的界限是 6）除外。
的情况，并且这个数字很容易减少到五种，但是将颜色数量一直减少到四种被证明非常困难。阿佩尔和哈肯（1977 年）最终获得了这个结果，他们构建了一个计算机辅助证明，证明四种颜色就足够了。然而，由于部分证明包括计算机对许多离散情况的详尽分析，一些数学家不接受它。但是，尚未发现任何缺陷，因此该证明看起来有效。罗伯逊等人。（1996 年；托马斯 1998 年）构建了一个更短、独立的证明。