论文标题
3个临界子图
3-critical subgraphs of snarks
论文作者
论文摘要
在本文中，我们进一步了解了众所周知的第二类立方图或蛇形的结构。我们通过调查他们的3个临界子图，或者按照我们所说的最小冲突子图来做到这一点。我们考虑Snark的最小冲突子图与可能的最小4边色有关。我们充分表征了蛇的阻力与一组最小冲突的子图之间的关系。也就是说，我们表明蛇形的电阻等于可以从snark中选择的最小边数，因此选择中至少包含每个最小冲突子图的边缘。我们类似地表征了我们所谓的\ textit {textit {关键子图}的关系与最小冲突的子图的集合之间的关系。关键的子图是所有边缘的集合，这些边缘在Snark的某些最小颜色中矛盾。除此之外，我们定义了最小冲突子图的组或\ textit {clusters}。然后，我们强调了一些有趣的属性和与最小冲突子图的集群有关的问题。
In this paper we further our understanding of the structure of class two cubic graphs, or snarks, as they are commonly known. We do this by investigating their 3-critical subgraphs, or as we will call them, minimal conflicting subgraphs. We consider how the minimal conflicting subgraphs of a snark relate to its possible minimal 4-edge-colourings. We fully characterise the relationship between the resistance of a snark and the set of minimal conflicting subgraphs. That is, we show that the resistance of a snark is equal to the minimum number of edges which can be selected from the snark, such that the selection contains at least one edge from each minimal conflicting subgraph. We similarly characterise the relationship between what we call \textit{the critical subgraph} of a snark and the set of minimal conflicting subgraphs. The critical subgraph being the set of all edges which are conflicting in some minimal colouring of the snark. Further to this, we define groups, or \textit{clusters}, of minimal conflicting subgraphs. We then highlight some interesting properties and problems relating to clusters of minimal conflicting subgraphs.
