# Proper Orientation Number of Triangle-free Bridgeless Outerplanar Graphs

## TL;DR

This paper investigates the proper orientation number of triangle-free outerplanar graphs, establishing new upper bounds for 2-connected and bridgeless cases, advancing understanding of graph orientations.

## Contribution

It provides improved upper bounds for the proper orientation number of triangle-free outerplanar graphs, specifically 3 for 2-connected and 4 for bridgeless cases.

## Key findings

- Proper orientation number is at most 3 for 2-connected triangle-free outerplanar graphs.
- Proper orientation number is at most 4 for triangle-free bridgeless outerplanar graphs.
- Addresses an open question about boundedness of proper orientation numbers in outerplanar graphs.

## Abstract

An orientation of $G$ is a digraph obtained from $G$ by replacing each edge by exactly one of two possible arcs with the same endpoints. We call an orientation \emph{proper} if neighbouring vertices have different in-degrees. The proper orientation number of a graph $G$, denoted by $\vec{\chi}(G)$, is the minimum maximum in-degree of a proper orientation of G. Araujo et al. (Theor. Comput. Sci. 639 (2016) 14--25) asked whether there is a constant $c$ such that $\vec{\chi}(G)\leq c$ for every outerplanar graph $G$ and showed that $\vec{\chi}(G)\leq 7$ for every cactus $G.$ We prove that $\vec{\chi}(G)\leq 3$ if $G$ is a triangle-free $2$-connected outerplanar graph and $\vec{\chi}(G)\leq 4$ if $G$ is a triangle-free bridgeless outerplanar graph.