Geodesic
A decomposition of Gallai multigraphs
Discussiones Mathematicae. Graph Theory, Tome 34 (2014) no. 2, pp. 331-352

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An edge-colored cycle is rainbow if its edges are colored with distinct colors. A Gallai (multi)graph is a simple, complete, edge-colored (multi)graph lacking rainbow triangles. As has been previously shown for Gallai graphs, we show that Gallai multigraphs admit a simple iterative construction. We then use this structure to prove Ramsey-type results within Gallai colorings. Moreover, we show that Gallai multigraphs give rise to a surprising and highly structured decomposition into directed trees
Keywords: edge coloring, Gallai multigraph 
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Halperin, Alexander; Magnant, Colton; Pula, Kyle. A decomposition of Gallai multigraphs. Discussiones Mathematicae. Graph Theory, Tome 34 (2014) no. 2, pp. 331-352. http://geodesic.mathdoc.fr/item/DMGT_2014_34_2_a8/