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 by Durocher and Mondal (SIAM J Discrete Math 32(4):2703–2719, 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.

一组彩色图形是兼容的，如果对于每种颜色i，每个图形中颜色i的顶点数量均相同。同时嵌入k个兼容颜色的图形（每个图形有n个顶点）由这些图形的k个平面多段线图组成，以便将相同颜色的顶点映射到一组通用的顶点位置。我们证明\（k \ in o（\ log \ log n）\）彩色平面图的同时嵌入（每个都有n个顶点）总是可以通过每个边缘的折线数为亚线性来计算。具体来说，我们在每个边的折弯数上显示\（O（\ min \ {c，n ^ {1-1 / \ gamma} \}）\）上限，其中\（\ gamma = 2 ^ {\ lceil k / 2 \ rceil} \），c是颜色的总数。我们的边界来自Durocher和Mondal（SIAM J Discrete Math 32（4）：2703–2719，2018）对先前已知算法的更好分析得出的结果，将k的边界以及折弯复杂度提高了一个因子的\（\ 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 \）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顶点平面图集，其中C 是颜色的数量。