Abstract

The vertex 3-colourability problem is to verify whether it is possible to split the vertex set of a given graph into three subsets of pairwise nonadjacent vertices or not. This problem is known to be NP-complete for planar graphs of the maximum face length at most 4 (and, even, additionally, of the maximum vertex degree at most 5), and it can be solved in linear time for planar triangulations. Additionally, the vertex 3-colourability problem is NP-complete for planar graphs of the maximum vertex degree at most 4, but it can be solved in constant time for graphs of the maximum vertex degree at most 3. It would be interesting to investigate classes of planar graphs with simultaneously bounded length of faces and the maximum vertex degree and to find the threshold, for which the complexity of the vertex 3-colourability problem is changed from polynomial-time solvability to NP-completeness. In this paper, we prove NP-completeness of the vertex 3-colourability problem for planar graphs of the maximum vertex degree at most 4, whose faces are of length no more than 5.

中文翻译：

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具有短面和有界最大顶点度的图的3-着色

摘要

顶点 3-可色性问题是验证是否可以将给定图的顶点集拆分为成对不相邻顶点的三个子集。对于最大面长度最多为 4（甚至最大顶点度数最多为 5）的平面图，这个问题已知是 NP 完全的，并且对于平面三角剖分可以在线性时间内解决。此外，顶点 3-着色性问题对于最大顶点度最多为 4 的平面图是 NP 完全的，但对于最大顶点度最多为 3 的图可以在恒定时间内解决。 研究类会很有趣具有同时有界面长度和最大顶点度的平面图并找到阈值，其中顶点 3-可着色性问题的复杂性从多项式时间可解性变为 NP-完全性。在本文中，我们证明了顶点度数最多为 4，面长度不超过 5 的平面图的顶点 3-可色性问题的 NP 完备性。