The 2-Ranking Numbers of Graphs

TL;DR

This paper investigates the 2-ranking number in graphs, establishing exact values for hypercubes, bounds for Cartesian products of cycles and complete graphs, and bounds for subcubic graphs and graphs with bounded degree.

Contribution

It provides exact and asymptotic bounds for the 2-ranking number in various classes of graphs, including hypercubes, Cartesian products, and graphs with bounded degree, advancing understanding of graph ranking properties.

Findings

2-ranking number of hypercube $Q_n$ is $n+1$

Bounds on 2-ranking number for Cartesian products $K_m imes K_n$

Maximum 2-ranking number for subcubic graphs is 7

Abstract

In a graph whose vertices are assigned integer ranks, a path is well-ranked if the endpoints have distinct ranks or some interior point has a higher rank than the endpoints. A ranking is an assignment of ranks such that all nontrivial paths are well-ranked. A $k$-ranking is a relaxation in which all nontrivial paths of length at most $k$ are well-ranked. The $k$-ranking number of a graph $G$ is the minimum $t$ such that there is a $k$-ranking of $G$ using ranks in $\{1,\ldots,t\}$. We prove that the $2$-ranking number of the $n$-dimensional hypercube $Q_n$ is $n+1$. As a corollary, we improve the bounds on the star chromatic number of products of cycles when each cycle has length divisible by $4$. For $m\le n$, we show that the $2$-ranking number of $K_m \mathop\square K_n$ is $\Omega(n\log m)$ and $O(nm^{\log_2(3)-1})$ with an asymptotic result when $m$ is constant and an exact result when $m!$ divides $n$. We prove that every subcubic graph has $2$-ranking number at most $7$, and we also prove the existence of a graph with maximum degree $k$ and $2$-ranking number $\Omega(k^2/\log(k))$.