Maximum minimal rankings of oriented trees

Abstract

Given a graph $G$, a $k$-ranking is a labeling of the vertices using $k$ labels so that every path between two vertices with the same label contains a vertex with a larger label. A $k$-ranking $f$ is minimal if for all $v\in V\left(G\right)$ we have $f\left(v\right)$ $\le g\left(v\right)$ for all rankings $g$. We explore this problem for directed graphs. Here every directed path between two vertices with the same label contains a vertex with a larger label. The rank number of a digraph $D$ is the smallest $k$ such that $D$ has a minimal $k$-ranking. The arank number of a digraph is the largest $k$ such that $D$ has a minimal $k$-ranking. We present new results involving rank numbers and arank numbers of directed graphs. In 1999, Kratochvíl and Tuza showed that the rank number of an oriented of a tree is bounded by one greater than the rank number of its longest directed path. We show that the arank analog does not hold. In fact we will show that the arank number of an oriented tree can be made arbitrarily large where the largest directed path has only three vertices.