Plummer and Toft conjectured in 1987 that the vertices of every 3-connected plane graph with maximum face size ${\Delta }^{\star }$ can be colored using at most ${\Delta }^{\star }+2$ colors in such a way that no face is incident with two vertices of the same color. The conjecture has been proven for ${\Delta }^{\star }=3$, ${\Delta }^{\star }=4$ and ${\Delta }^{\star }\ge 18$. We prove the conjecture for ${\Delta }^{\star }=16$ and ${\Delta }^{\star }=17$.

中文翻译：

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最大面大小为16和17的平面图的循环着色

1987年Plummer和Toft推测，每3个相连平面图的顶点具有最大的脸部尺寸 ${\Delta }^{\star }$ 最多可以使用 ${\Delta }^{\star }+2$以相同的颜色使两个面都没有面入射的方式着色。这个猜想已经证明了${\Delta }^{\star }=3$， ${\Delta }^{\star }=4$ 和 ${\Delta }^{\star }\ge \mathrm{18岁}$。我们证明了${\Delta }^{\star }=16$ 和 ${\Delta }^{\star }=17$。