Taiwanese Journal of Mathematics

Injective Chromatic Number of Outerplanar Graphs

Mahsa Mozafari-Nia and Behnaz Omoomi

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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

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

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

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Zentralblatt MATH identifier

Primary: 05C15: Coloring of graphs and hypergraphs

injective coloring injective chromatic number outerplanar graph


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

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