Applications
Applications of fractional graph coloring include activity scheduling. In this case, the graph G is a conflict graph: an edge in G between the nodes u and v denotes that u and v cannot be active simultaneously. Put otherwise, the set of nodes that are active simultaneously must be an independent set in graph G.
An optimal fractional graph coloring in G then provides a shortest possible schedule, such that each node is active for (at least) 1 time unit in total, and at any point in time the set of active nodes is an independent set. If we have a solution x to the above linear program, we simply traverse all independent sets I in an arbitrary order. For each I, we let the nodes in I be active for time units; meanwhile, each node not in I is inactive.
In more concrete terms, each node of G might represent a radio transmission in a wireless communication network; the edges of G represent interference between radio transmissions. Each radio transmission needs to be active for 1 time unit in total; an optimal fractional graph coloring provides a minimum-length schedule (or, equivalently, a maximum-bandwidth schedule) that is conflict-free.
Read more about this topic: Fractional Coloring