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3. We also give an upper bound for the oriented chromatic number of planar graphs with girth at least 11.
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4.
doi:10.1016/j.ipl.2006.09.007 
Copyright © 2006 Elsevier B.V. All rights reserved.
Oriented colorings of 2-outerplanar graphs
Received 20 January 2006;
revised 12 September 2006;
accepted 15 September 2006.
Communicated by L. Boasson.
Available online 23 October 2006.
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Abstract
A graph G is 2-outerplanar if it has a planar embedding such that the subgraph obtained by removing the vertices of the outer face is outerplanar. The oriented chromatic number of an oriented graph H is defined as the minimum order of an oriented graph H′ such that H has a homomorphism to H′. In this paper, we prove that 2-outerplanar graphs are 4-degenerate. We also show that oriented 2-outerplanar graphs have a homomorphism to the Paley tournament QR67, which implies that their (strong) oriented chromatic number is at most 67.
Keywords: Combinatorial problems; Oriented coloring; 2-outerplanar graphs
