Oriented Coloring of Triangle-Free Planar Graphs and 2-Outerplanar Graphs

Pascal Ochem 1 Alexandre Pinlou 2
2 ALGCO - Algorithmes, Graphes et Combinatoire
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
Abstract : A graph is planar if it can be embedded on the plane without edge-crossing. A graph is 2-outerplanar if it has a planar embedding such that the subgraph obtained by removing the vertices of the external face is outerplanar (i.e. with all its vertices on the external face). An oriented k-coloring of an oriented graph G is a homomorphism from G to an oriented graph H of order k. We prove that every oriented triangle-free planar graph has an oriented chromatic number at most 40, that improves the previous known bound of 47 due to Borodin and Ivanova [Borodin, O. V. and Ivanova, A. O., An oriented colouring of planar graphs with girth at least 4, Sib. Electron. Math. Reports, vol. 2, 239-249, 2005]. We also prove that every oriented 2-outerplanar graph has an oriented chromatic number at most 40, that improves the previous known bound of 67 due to Esperet and Ochem [Esperet, L. and Ochem, P. Oriented colouring of 2-outerplanar graphs, Inform. Process. Lett., vol. 101(5), 215-219, 2005].
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Pascal Ochem, Alexandre Pinlou. Oriented Coloring of Triangle-Free Planar Graphs and 2-Outerplanar Graphs. LAGOS'11: VI Latin-American Algorithms, Graphs and Optimization Symposium, Mar 2011, Bariloche, Argentina. pp.123-128, ⟨10.1016/j.endm.2011.05.022⟩. ⟨lirmm-00530543⟩

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