A set of colored graphs are compatible, if for every color $i$, the number of vertices of color $i$ is the same in every graph. A simultaneous embedding of $k$ compatibly colored graphs, each with $n$ vertices, consists of $k$ planar polyline drawings of these graphs such that the vertices of the same color are mapped to a common set of vertex locations. We prove that simultaneous embedding of $k\in o(\log \log n)$ colored planar graphs, each with $n$ vertices, can always be computed with a sublinear number of bends per edge. Specifically, we show an $O(\min\{c, n^{1-1/\gamma}\})$ upper bound on the number of bends per edge, where $\gamma = 2^{\lceil k/2 \rceil}$ and $c$ is the total number of colors. Our bound, which results from a better analysis of a previously known algorithm [Durocher and Mondal, SIAM J. Discrete Math., 32(4), 2018], improves the bound for $k$, as well as the bend complexity by a factor of $\sqrt{2}^{k}$. The algorithm can be generalized to obtain small universal point sets for colored graphs. We prove that $n\lceil c/b \rceil$ vertex locations, where $b\ge 1$, suffice to embed any set of compatibly colored $n$-vertex planar graphs with bend complexity $O(b)$, where $c$ is the number of colors.

中文翻译：

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彩色图的同时嵌入

一组彩色图形是兼容的，如果对于每个颜色$ i $，每个图形中颜色$ i $的顶点数量相同。同时嵌入每个具有$ n $个顶点的$ k $彩色着色图，由这些图的$ k $平面多段线图组成，以便将相同颜色的顶点映射到一组相同的顶点位置。我们证明，同时嵌入$ k \ o（\ log \ log n）$彩色平面图（每个都有$ n $个顶点），总是可以使用每个边缘的折线亚线性数来计算。具体来说，我们在每个边的折弯数量上显示$ O（\ min \ {c，n ^ {1-1 / \ gamma} \}）$上限，其中$ \ gamma = 2 ^ {\ lceil k / 2 \ rceil} $和$ c $是颜色的总数。我们的界限来自对先前已知算法的更好分析[Durocher and Mondal，SIAM J.Discrete Math。，32（4），2018]，将$ k $的边界和折弯复杂度提高了$ \ sqrt {2} ^ {k} $。可以对该算法进行一般化以获得彩色图的通用小点集。我们证明$ n \ lceil c / b \ rceil $个顶点位置，其中$ b \ ge 1 $足以嵌入任何具有弯曲复杂度$ O（b）$的颜色兼容的$ n $-顶点平面图，其中$ c $是颜色数。