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

Abstract

Given a weighted graph G x , where ( x ( v ) : v ∈ V ) is a non-negative, real-valued weight assigned to the vertices of G , let B ( G x ) be an upper bound on the fractional chromatic number of the weighted graph G x ; so χ 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 β ( G ) = sup x ≠ 0 B ( G x ) χ f ( G x ) . This graph invariant is shown to be equal to the size of the largest star subgraph in the graph. These results have implications for the design and performance of distributed communication networks.