Path coloring

In graph theory, path coloring usually refers to one of two problems: In both the above problems, the goal is usually to minimise the number of colors used in the coloring. In different variants of path coloring, $$G$$ may be a simple graph, digraph or multigraph.
 * The problem of coloring a (multi)set of paths $$R$$ in graph $$G$$, in such a way that any two paths of $$R$$ which share an edge in $$G$$ receive different colors. Set $$R$$ and graph $$G$$ are provided at input. This formulation is equivalent to vertex coloring the conflict graph of set $$R$$, i.e. a graph with vertex set $$R$$ and edges connecting all pairs of paths of $$R$$ which are not edge-disjoint with respect to $$G$$.
 * The problem of coloring (in accordance with the above definition) any chosen (multi)set $$R$$ of paths in $$G$$, such that the set of pairs of end-vertices of paths from $$R$$ is equal to some set or multiset $$I$$, called a set of requests. Set $$I$$ and graph $$G$$ are provided at input. This problem is a special case of a more general class of graph routing problems, known as call scheduling.