## Involve: A Journal of Mathematics

• Involve
• Volume 2, Number 3 (2009), 289-295.

### 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 $v∈V(G)$ we have $f(v)$ $≤g(v)$ 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.

#### Article information

Source
Involve, Volume 2, Number 3 (2009), 289-295.

Dates
Revised: 29 April 2009
Accepted: 3 June 2009
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.involve/1513799163

Digital Object Identifier
doi:10.2140/involve.2009.2.289

Mathematical Reviews number (MathSciNet)
MR2551126

Zentralblatt MATH identifier
1177.05044

#### Citation

Novotny, Sarah; Ortiz, Juan; Narayan, Darren. Maximum minimal rankings of oriented trees. Involve 2 (2009), no. 3, 289--295. doi:10.2140/involve.2009.2.289. https://projecteuclid.org/euclid.involve/1513799163

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