We introduce a rainbow variant of separating path systems. For up to four colors, we investigate the minimum size of such a separating system for various graph classes such as paths, cycles, and trees. Furthermore, we analyze the behavior for any number of colors, establishing bounds for a wide class including complete graphs and Erdős-Rényi random graphs.

We prove that for any two graphs G and H , the edges of G can be strongly separated by a collection of linearly many subdivisions of H and single edges. This confirms a conjecture of Botler and Naia.

We establish minimum and maximum degree bounds for digraphs that ensure the containment of oriented trees of smaller order. We prove that, asymptotically, every large digraph of minimum semidegree above 2k/3 having vertices of out-degree and in-degree above k contains each large balanced antidirected bounded-degree tree with k arcs. Also, we show that, asymptotically, every large digraph of minimum semidegree above 3k/5 having vertices of out-degree and in-degree above 2k contains each large balanced antidirected bounded-degree tree with k arcs. Our result is restricted to balanced antidirected trees of bounded degree but may hold for other oriented trees as well.