# Injective chromatic number of outerplanar graphs

## TL;DR

This paper investigates the injective chromatic number of outerplanar graphs, establishing tight bounds based on maximum degree and girth, and providing exact values for specific subclasses.

## Contribution

It derives new tight bounds for the injective chromatic number of outerplanar graphs considering maximum degree and girth, including exact values for certain cases.

## Key findings

- For all outerplanar graphs, hi_i(G) st; bound is tight.
- For elta=3, hi_i(G) st+1; tight for girth 3 and 4.
- For 2-connected outerplanar graphs with elta=3, g , and elta , g , hi_i(G) = elta.

## Abstract

An injective coloring of a graph is a vertex coloring where two vertices with common neighbor receive distinct colors. The minimum integer $k$ that $G$ has a $k-$injective coloring is called injective chromatic number of $G$ and denoted by $\chi_i(G)$. In this paper, the injective chromatic number of outerplanar graphs with maximum degree $\Delta$ and girth $g$ is studied. It is shown that for every outerplanar graph, $\chi_i(G)\leq \Delta+2$, and this bound is tight. Then, it is proved that for outerplanar graphs with $\Delta=3$, $\chi_i(G)\leq \Delta+1$ and the bound is tight for outerplanar graphs of girth three and $4$. Finally, it is proved that, the injective chromatic number of $2-$connected outerplanar graphs with $\Delta=3$, $g\geq 6$ and $\Delta\geq 4$, $g\geq 4$ is equal to $\Delta$.