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.

An ( n , m ) -graph is characterized by n types of arcs and m types of edges. A homomorphism of an ( n , m ) -graph G to an ( n , m ) -graph H , is a vertex mapping that preserves adjacency, direction, and type. The ( n , m ) -chromatic number of G , denoted by χ n , m ( G ) , is the minimum value of | V ( H ) | such that there exists a homomorphism of G to H . The theory of homomorphisms of ( n , m ) -graphs have connections with graph theoretic concepts like harmonious coloring, nowhere-zero flows; with other mathematical topics like binary predicate logic , Coxeter groups; and has application to the Query Evaluation Problem (QEP) in graph database. In this article, we show that the arboricity of G is bounded by a function of χ n , m ( G ) but not the other way around. Additionally, we show that the acyclic chromatic number of G is bounded by a function of χ n , m ( G ) , a result already known in the reverse direction. Furthermore, we prove that the ( n , m ) -chromatic number for the family of graphs with maximum average degree less than 2 + 2 4 ( 2 n + m ) − 1 , including the subfamily of planar graphs with girth at least 8 ( 2 n + m ) , equals 2 ( 2 n + m ) + 1 . This improves upon previous findings, which proved the ( n , m ) -chromatic number for planar graphs with girth at least 10 ( 2 n + m ) − 4 is 2 ( 2 n + m ) + 1 . It is established that the ( n , m ) -chromatic number for the family T 2 of partial 2-trees is both bounded below and above by quadratic functions of ( 2 n + m ) , with the lower bound being tight when ( 2 n + m ) = 2 . We prove 14 ≤ χ ( 0 , 3 ) ( T 2 ) ≤ 15 and 14 ≤ χ ( 1 , 1 ) ( T 2 ) ≤ 21 which improves both known lower bounds and the former upper bound. Moreover, for the latter upper bound, to the best of our knowledge we provide the first theoretical proof.