Tilings in graphons

TL;DR

This paper extends the concept of graph tilings to graphons, establishing a transference principle between finite graphs and their limits, and applies this to random graphs and classical tiling theorems.

Contribution

It introduces a graphon-based framework for tilings, including LP-duality and property testing connections, and applies it to inhomogeneous random graphs and classical tiling theorems.

Findings

Established a transference principle between finite graphs and graphons.

Determined the asymptotic F-tiling number in inhomogeneous random graphs.

Provided a new proof of a strengthened Komlos tiling theorem.

Abstract

We introduce a counterpart to the notion of vertex disjoint tilings by copy of a fixed graph F to the setting of graphons. The case F=K_2 gives the notion of matchings in graphons. We give a transference statement that allows us to switch between the finite and limit notion, and derive several favorable properties, including the LP-duality counterpart to the classical relation between the fractional vertex covers and fractional matchings/tilings, and discuss connections with property testing. As an application of our theory, we determine the asymptotically almost sure F-tiling number of inhomogeneous random graphs \mathbb{G}(n,W). As another application, in an accompanying paper [Hladky, Hu, Piguet: Komlos's tiling theorem via graphon covers, preprint] we give a proof of a strengthening of a theorem of Komlos [Komlos: Tiling Tur\'an Theorems, Combinatorica, 2000].