Edge‐colored complete graphs without properly colored even cycles: A full characterization

The structure of edge-colored complete graphs containing no properly colored triangles has been characterized by Gallai back in the 1960s. More recently, Cǎda et al. and Fujita et al. independently determined the structure of edge-colored complete bipartite graphs containing no properly colored (Formula presented.). We characterize the structure of edge-colored complete graphs containing no properly colored even cycles, or equivalently, without a properly colored (Formula presented.) or (Formula presented.). In particular, we first deal with the simple case of 2-edge-colored complete graphs, using a result of Yeo. Next, for (Formula presented.), we define four classes of (Formula presented.) -edge-colored complete graphs without properly colored even cycles and prove that any (Formula presented.) -edge-colored complete graph without a properly colored even cycle belongs to one of these four classes.