Independent sets, cliques, and colorings in graphons

TL;DR

This paper explores graphon analogs of classical graph parameters like chromatic and clique numbers, establishing properties, dualities, and characterizations of perfect graphons in the continuous graph limit setting.

Contribution

It introduces and analyzes graphon versions of key graph invariants, providing foundational properties and characterizations for these continuous graph models.

Findings

Established properties of the independence set polytope in graphons

Proved duality between fractional chromatic and clique numbers in graphons

Characterized perfect graphons via densities of odd cycles and their complements

Abstract

We study graphon counterparts of the chromatic and the clique number, the fractional chromatic number, the b-chromatic number, and the fractional clique number. We establish some basic properties of the independence set polytope in the graphon setting, and duality properties between the fractional chromatic number and the fractional clique number. We present a notion of perfect graphons and characterize them in terms of induced densities of odd cycles and its complements.