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Injective chromatic number of outerplanar graphs
- Publication Year :
- 2017
-
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 $\Delta$.<br />Comment: 13 pages, 6 figures
- Subjects :
- Mathematics - Combinatorics
05C10
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1706.02335
- Document Type :
- Working Paper