The performance of an upper bound on the fractional chromatic number of weighted graphs

Given a weighted graph G(x), where (x(upsilon) : upsilon is an element of V) is a non-negative, real-valued weight assigned to the vertices of G, let B(G) be an upper bound on the fractional chromatic number of the weighted graph G(x); so chi(f)(G(x)) <= B(G(x)). We consider a particular upper bound B resulting from a generalization of the greedy coloring algorithm to weighted graphs. To investigate the worst-case performance of this upper bound, we study the graph invariant