Discussiones Mathematicae Graph Theory
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.
Halperin, Alexander, Colton Magnant, Kyle Pula.
"A Decomposition of Gallai Multigraphs."
Discussiones Mathematicae Graph Theory, 34: 331-352.