On some upper bounds on the fractional chromatic number of weighted graphs

Given a weighted graph Gx, where (x(v) : v ∈ V ) is a non-negative, realvalued weight assigned to the vertices of G, let B(Gx) be an upper bound on the fractional chromatic number of the weighted graph Gx; so χf (Gx) ≤ B(Gx). To investigate the worst-case performance of the upper bound B, we study the graph invariant β(G) = sup x 6=0 B(Gx) χf (Gx) . In recent work a particular upper bound resulting from the generalization of the greedy coloring algorithm was considered and the corresponding graph invariant was studied. In this work, we study some stronger upper bounds on the fractional chromatic number and the corresponding graph invariants. We derive some bounds for these graph invariants and obtain some explicit expressions for some families of graphs.