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December, 2018 Injective Chromatic Number of Outerplanar Graphs
Mahsa Mozafari-Nia, Behnaz Omoomi
Taiwanese J. Math. 22(6): 1309-1320 (December, 2018). DOI: 10.11650/tjm/180807


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 $\Delta$.


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Mahsa Mozafari-Nia. Behnaz Omoomi. "Injective Chromatic Number of Outerplanar Graphs." Taiwanese J. Math. 22 (6) 1309 - 1320, December, 2018.


Received: 30 September 2017; Revised: 3 April 2018; Accepted: 13 August 2018; Published: December, 2018
First available in Project Euclid: 20 August 2018

zbMATH: 07021691
MathSciNet: MR3880234
Digital Object Identifier: 10.11650/tjm/180807

Primary: 05C15

Keywords: injective chromatic number , injective coloring , outerplanar graph

Rights: Copyright © 2018 The Mathematical Society of the Republic of China


Vol.22 • No. 6 • December, 2018
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