The 2-Ranking Numbers of Graphs

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 $\Omega(k^2/\log(k))$.