关于图的顶点排序及其相关问题
On Vertex Rankings of Graphs and its Relatives
摘要：
一个图的顶点排序是给图的顶点分配秩（或颜色），以这种方式，任何连接两个相等秩的顶点的简单路径必须包含一个更高秩的顶点。在本文中，我们研究了这个概念的放松，其中上述要求应该只适用于一些固定的$l$的有界长度$l$的道路。例如，案例$l=2$已经表现出与正确着色完全不同的行为。我们证明了树、可平面图、不含固定子图和退化图等图族所需的最小秩数的上界和下界。
英文摘要：
A vertex ranking of a graph is an assignment of ranks (or colors) to the vertices of the graph, in such a way that any simple path connecting two vertices of equal rank, must contain a vertex of a higher rank. In this paper we study a relaxation of this notion, in which the requirement above should only hold for paths of some bounded length $l$ for some fixed $l$. For instance, already the case $l=2$ exhibit quite a different behavior than proper coloring. We prove upper and lower bounds on the minimum number of ranks required for several graph families, such as trees, planar graphs, graphs excluding a fixed minor and degenerate graphs.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
