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

TL;DR

This paper investigates upper bounds on the fractional chromatic number of weighted graphs, analyzing their worst-case performance and relating them to graph invariants like star subgraph size, with applications in distributed systems and wireless networks.

Contribution

It introduces the invariant β(G) to measure the worst-case ratio of upper bounds to the fractional chromatic number, linking it to graph structure and network resource estimation.

Findings

β(G) relates to the size of the largest star subgraph in some cases

The invariant provides insights into the efficiency of upper bounds in resource allocation

Results have implications for decentralized network design

Abstract

Given a weighted graph $G_\bx$, where $(x(v): v \in V)$ is a non-negative, real-valued weight assigned to the vertices of G, let $B(G_\bx)$ be an upper bound on the fractional chromatic number of the weighted graph $G_\bx$; so $\chi_f(G_\bx) \le B(G_\bx)$. To investigate the worst-case performance of the upper bound $B$, we study the graph invariant $$\beta(G) = \sup_{\bx \ne 0} \frac{B(G_\bx)}{\chi_f(G_\bx)}.$$ \noindent This invariant is examined for various upper bounds $B$ on the fractional chromatic number. In some important cases, this graph invariant is shown to be related to the size of the largest star subgraph in the graph. This problem arises in the area of resource estimation in distributed systems and wireless networks; the results presented here have implications on the design and performance of decentralized communication networks.