2016/09/20 by Ilan Karpas, Ofer Neiman, Karpas, Ilan +3
Computer Science · Mathematics · #Advanced Graph Theory Research #Limits and Structures in Graph Theory #Complexity and Algorithms in Graphs
paper · pdf · doi:10.48550/arxiv.1609.06111
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.