Perfect graphs and norms

Abstract

For a given undirected graphG=(V, Г) withn vertices we define four norms on ℝⁿ, namely

, where ℰ (resp.

) stands for the family of all maximal cliques inG (resp.$\tilde G$, the complement ofG). The goal of this note is to demonstrate the usefulness of some notions and techniques from functional analysis in graph theory by showing how Theorem 2.1 (G is ℒ-perfect if and only if the norms

are equal) together with the reflexivity of the space ℝⁿ equipped with either of the norms above easily yield one new result (Theorem 2.2) and two known characterizations of perfect graphs (Theorems 2.3–2.4). Namely, Theorem 2.2 provides a characterization of ℒ-perfection that is strongly related to that of Lovász (1972). It implies that the Lovász inequality is exactly the classical Schwartz inequality for the space (ℝⁿ, ∥·∥^ℰ) restricted to (0, 1) vectorsx, y satisfyingx = y. Theorem 2.3 is well known as the Perfect Graph Theorem, while Theorem 2.4, due to V. Chvátal and D.R. Fulkerson, characterizes ℒ-perfection of a graphG in terms of the equality between the vertex packing polytope ofG and the fractional vertex packing polytope ofG.