Geodesic
Decompositions of Plane Graphs Under Parity Constrains Given by Faces
Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 3, pp. 521-530

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An edge coloring of a plane graph G is facially proper if no two faceadjacent edges of G receive the same color. A facial (facially proper) parity edge coloring of a plane graph G is an (facially proper) edge coloring with the property that, for each color c and each face f of G, either an odd number of edges incident with f is colored with c, or color c does not occur on the edges of f. In this paper we deal with the following question: For which integers k does there exist a facial (facially proper) parity edge coloring of a plane graph G with exactly k colors?
Keywords: plane graph, parity partition, edge coloring 
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Czap, Július; Tuza, Zsolt. Decompositions of Plane Graphs Under Parity Constrains Given by Faces. Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 3, pp. 521-530. http://geodesic.mathdoc.fr/item/DMGT_2013_33_3_a3/