Properly colored spanning trees via subdivision of a given tree in monochromatic triangle-free edge-colored complete graphs

An edge-colored graph is termed properly colored if any two adjacent edges are assigned distinct colors and monochromatic if all its edges share the same color. An edge-colored graph is called mono- H -free if it contains no monochromatic copy of H . We observe that if every mono- H -free edge-colored complete graph contains a properly colored Hamilton path, then H must be a star or a triangle. Motivated by this observation, we investigate the existence of properly colored spanning trees under the mono-C₃-free condition. For a given tree T₀, let T(n,T₀) denote the set of all n -vertex trees that are subdivisions of T₀. In general, we conjecture that for any tree T₀, every mono-C₃-free edge-colored complete graph K_n with n sufficiently large relative to |V(T₀)| contains a properly colored copy of every T∈T(n,T₀). We approach this conjecture from multiple angles. First, we show that for any tree T₀ with k edges, every mono-C₃-free edge-colored K_n with n≥(k+2)! contains a properly colored copy of some T∈T(n,T₀). Second, for the star S_k and every T∈T(n,S_k), we demonstrate that every mono-C₃-free edge-colored K_n with n≥(k+3)! contains a properly colored copy of T . Third, we verify the conjecture for a specific class of host edge-colored complete graphs explicitly constructed on {0,1}^k. The proofs utilize several newly introduced absorbing structures and techniques from a specific class of multipartite tournaments. Denote by g(S_k,C₃) the maximum number N such that there exists an edge-colored K_N containing neither a rainbow S_k nor a monochromatic C₃. We further note that the lower bounds on n in the first two results can be improved as any significant advance is made on the upper bound of g(S_k,C₃).