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Asymptotic density of graphs excluding disconnected minors
Journal of Combinatorial Theory Series B ( IF 1.2 ) Pub Date : 2020-09-23 , DOI: 10.1016/j.jctb.2020.09.007 Rohan Kapadia , Sergey Norin , Yingjie Qian
中文翻译:
图的渐近密度,不包括未连接的未成年人
更新日期:2020-09-23
Journal of Combinatorial Theory Series B ( IF 1.2 ) Pub Date : 2020-09-23 , DOI: 10.1016/j.jctb.2020.09.007 Rohan Kapadia , Sergey Norin , Yingjie Qian
For a graph H, let where the maximum is taken over all graphs G on n vertices not containing H as a minor. Thus is the asymptotic maximum density of graphs not containing H as a minor. Employing a structural lemma due to Eppstein, we prove new upper bounds on for disconnected graphs H. In particular, we determine whenever H is a union of cycles. Finally, we investigate the behaviour of for fixed r, where denotes the union of s disjoint copies of the complete graph on r vertices. Improving on a result of Thomason, we show that and
中文翻译:
图的渐近密度,不包括未连接的未成年人
对于图H,让其中在不包含H作为次要元素的n个顶点上的所有图G上取最大值。从而是不含次要元素H的图的渐近最大密度。利用因普斯坦的结构引理,我们证明了新的上界断开连接图^ h。特别是,我们确定每当H是周期的并集时。最后,我们调查了对于固定r,其中表示在r个顶点上完整图的s个不相交副本的并集。通过对Thomason的改进,我们表明 和