The Electronic Journal of Combinatorics
For a fixed integer m, we consider edge colorings of complete graphs which contain no properly edge colored cycle Cm as a subgraph. Within colorings free of these subgraphs, we establish global structure by bounding the number of colors that can induce a spanning and connected subgraph. In the case of smaller cycles, namely C4,C5, and C6, we show that our bounds are sharp.
Coll, Vincent, Colton Magnant, Kathleen Ryan.
"Structure of Colored Complete Graphs Free of Proper Cycles."
The Electronic Journal of Combinatorics, 19 (4): 1-16.