On Vertex Rankings of Graphs and its Relatives

TL;DR

This paper explores a relaxed vertex ranking concept in graphs where the rank condition applies only to paths of bounded length, providing bounds for various graph families.

Contribution

It introduces a bounded-length path relaxation of vertex ranking and establishes bounds for different classes of graphs, expanding understanding of graph coloring variants.

Findings

Bounds established for trees, planar graphs, and minor-excluding graphs.

Different behaviors observed for the case l=2 compared to proper coloring.

Provides theoretical limits on the number of ranks needed for various graph families.

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 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.