## Taiwanese Journal of Mathematics

- Taiwanese J. Math.
- Volume 22, Number 6 (2018), 1309-1320.

### Injective Chromatic Number of Outerplanar Graphs

Mahsa Mozafari-Nia and Behnaz Omoomi

#### Abstract

An injective coloring of a graph is a vertex coloring where two vertices with common neighbor receive distinct colors. The minimum integer $k$ such 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 every outerplanar graph $G$ has $\chi_i(G) \leq \Delta+2$, and this bound is tight. Then, it is proved that for an outerplanar graph $G$ with $\Delta = 3$, $\chi_i(G) \leq \Delta+1$ and the bound is tight for outerplanar graphs of girth $3$ 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 $\Delta$.

#### Article information

**Source**

Taiwanese J. Math., Volume 22, Number 6 (2018), 1309-1320.

**Dates**

Received: 30 September 2017

Revised: 3 April 2018

Accepted: 13 August 2018

First available in Project Euclid: 20 August 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.twjm/1534730418

**Digital Object Identifier**

doi:10.11650/tjm/180807

**Mathematical Reviews number (MathSciNet)**

MR3880234

**Zentralblatt MATH identifier**

07021691

**Subjects**

Primary: 05C15: Coloring of graphs and hypergraphs

**Keywords**

injective coloring injective chromatic number outerplanar graph

#### Citation

Mozafari-Nia, Mahsa; Omoomi, Behnaz. Injective Chromatic Number of Outerplanar Graphs. Taiwanese J. Math. 22 (2018), no. 6, 1309--1320. doi:10.11650/tjm/180807. https://projecteuclid.org/euclid.twjm/1534730418