Geodesic
Triangle Decompositions of Planar Graphs
Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 3, pp. 643-659

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A multigraph G is triangle decomposable if its edge set can be partitioned into subsets, each of which induces a triangle of G, and rationally triangle decomposable if its triangles can be assigned rational weights such that for each edge e of G, the sum of the weights of the triangles that contain e equals 1. We present a necessary and sufficient condition for a planar multigraph to be triangle decomposable. We also show that if a simple planar graph is rationally triangle decomposable, then it has such a decomposition using only weights 0, 1 and 1/2 . This result provides a characterization of rationally triangle decomposable simple planar graphs. Finally, if G is a multigraph with K4 as underlying graph, we give necessary and sufficient conditions on the multiplicities of its edges for G to be triangle and rationally triangle decomposable.
Keywords: planar graphs, triangle decompositions, rational triangle decompositions 
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     title = {Triangle {Decompositions} of {Planar} {Graphs}},
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Mynhardt, Christina M.; Bommel, Christopher M. van. Triangle Decompositions of Planar Graphs. Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 3, pp. 643-659. http://geodesic.mathdoc.fr/item/DMGT_2016_36_3_a9/