Title: On Vertex Rankings of Graphs and its Relatives
Authors: Ilan Karpas, Ofer Neiman and Shakhar Smorodinsky.
Abstract: 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 path connecting two vertices of equal rank, must contain a vertex of a higher rank.
In this paper we study a restriction of this notion, in which the requirement above should only hold for paths
of length at most $l$. For instance, already the case $l=2$ exhibit quite a different behavior than proper coloring.
We prove upper and lower bounds on the minimal number of ranks required for several graph families, such as trees,
planar graphs, graphs excluding a fixed minor and degenerate graphs.