Ban-Linial猜想与树型snarks
Ban--Linial's Conjecture and treelike snarks
摘要：
一个无桥三次图$G$称为有2-二分图，如果存在$G$的2-顶点着色（不一定是真的）使得：（i）色类具有相同的基数，（ii）单色分量是孤立的顶点或边。在2016年，Ban和Linial证明了除了著名的Petersen图之外，每个无桥立方图都允许2-二分。在同一文件中，它表明，每一类I无桥立方图承认这样的二分法。在图论中，第二类无桥三次图被称为Snark，特别是那些过度指数至少为5的图，即其边集不能被四个完美匹配覆盖的图。此外，在[J.图论，86（2）（2017），149- 158]中，Esperet等人指出，Ban-Linial猜想的可能反例必须具有至少5的循环流数。同一作者还指出，虽然经验证据表明，从彼得森图获得的几个图承认2-二分法，但它们不能提供任何一般证明。尽管有一些零星的计算结果，到目前为止，还没有一个普遍的结果已经证明了过度的指数和循环流数至少为5。在这项工作中，我们表明，树状的咆哮，这是一个无限的家庭的咆哮严重依赖于彼得森图和他们的循环流数和过度指数至少为5，承认2-二分法。
英文摘要：
A bridgeless cubic graph $G$ is said to have a 2-bisection if there exists a 2-vertex-colouring of $G$ (not necessarily proper) such that: (i) the colour classes have the same cardinality, and (ii) the monochromatic components are either an isolated vertex or an edge. In 2016, Ban and Linial conjectured that every bridgeless cubic graph, apart from the well-known Petersen graph, admits a 2-bisection. In the same paper it was shown that every Class I bridgeless cubic graph admits such a bisection. The Class II bridgeless cubic graphs which are critical to many conjectures in graph theory are known as snarks, in particular, those with excessive index at least 5, that is, whose edge set cannot be covered by four perfect matchings. Moreover, in [J. Graph Theory, 86(2) (2017), 149--158], Esperet et al. state that a possible counterexample to Ban--Linial's Conjecture must have circular flow number at least 5. The same authors also state that although empirical evidence shows that several graphs obtained from the Petersen graph admit a 2-bisection, they can offer nothing in the direction of a general proof. Despite some sporadic computational results, until now, no general result about snarks having excessive index and circular flow number both at least 5 has been proven. In this work we show that treelike snarks, which are an infinite family of snarks heavily depending on the Petersen graph and with both their circular flow number and excessive index at least 5, admit a 2-bisection.
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