Structure of colored complete graphs free of proper cycles (Q1953342)

Summary: For a fixed integer \(m\), we consider edge colorings of complete graphs which contain no properly edge colored cycle \(C_{m}\) 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 \(C_4,C_5\), and \(C_6\), we show that our bounds are sharp.