Properly colored short cycles in edge-colored graphs

Properly colored cycles in edge-colored graphs are closely related to directed oriented graphs. As an analogy of the well-known Caccetta–Häggkvist Conjecture, we study existence properly bounded length graph. We first prove that for all integers s and t with ≥ 2 , every graph G no K contains a spanning subgraph H which admits orientation D such cycle is . Using this result, show r 4 if Conjecture holds then order n minimum color degree at least / + 1 most In addition, also obtain asymptotically tight total condition ensures (or rainbow)