A weak coloring of a graph G embedded on a surface is a vertex coloring of G such that no face is monochromatic. The weak chromatic number of G is the minimum number k such that G has a weak k-coloring. Kündgen and Ramamurthi (J Combin Theory Ser B 85, 307–337, 2002) conjectured that for each positive integer k, there is a graph that has two different embeddings on the same surface whose weak chromatic numbers differ by at least k. In this paper, we answer this conjecture affirmatively in two ways.

中文翻译：

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图嵌入到具有不同弱色数的曲面中

的曲线图的一个弱着色ģ嵌入的表面上是一个顶点着色ģ使得没有脸是单色的。G的弱色数是最小值k，因此G具有弱的k色。Kündgen和Ramamurthi（J Combin Theory系列B 85，307–337，2002）推测，对于每个正整数k，存在一个图，该图在同一表面上具有两个不同的嵌入，其弱色数至少相差k。在本文中，我们以两种方式肯定地回答这个猜想。