Komlós's tiling theorem via graphon covers

Abstract

Komlós [Komlós: Tiling Turán Theorems, Combinatorica, 2000] determined the asymptotically optimal minimum‐degree condition for covering a given proportion of vertices of a host graph by vertex‐disjoint copies of a fixed graph H, thus essentially extending the Hajnal–Szemerédi theorem that deals with the case when H is a clique. We give a proof of a graphon version of Komlós's theorem. To prove this graphon version, and also to deduce from it the original statement about finite graphs, we use the machinery introduced in [Hladký, Hu, Piguet: Tilings in graphons, arXiv:1606.03113]. We further prove a stability version of Komlós's theorem.

Matching Polytons

“…Here, we extend the statement by allowing the sets in the sequence to be only "asymptotically independent". The proof is however similar to that in [6].…”

“…The way of deriving Theorem 20 from Theorem 19 is standard, and we refer the reader to [6] where this was done in detail in the context of a tiling theorem of Komlós, [9], which is a statement of a similar flavor.…”

Tilings in graphons

“…In Section 5 we determine the F-tiling number of inhomogeneous random graphs. In an accompanying paper [9] we give another application of the theory by proving a strengthened version of a theorem of Komlós [10] regarding tilings in finite graphs.…”

“…By combining Theorem 3.4 and Theorem 3.16 it suffices to show that no graphon arising as a limit of graphs from G has a fractional F-cover of size less than γ. We shall see one particular application of this scheme in Section 5 and another one is use in the accompanying paper [9] on Komlós's Theorem. Remark 3.17.…”

First steps in combinatorial optimization on graphons: Matchings

“…While this announcement is rather dry due to space constraints, we believe that it will convince the reader that many interesting combinatorial phenomena about matchings extend in a nontrivial way to the graphon setting. Also, note that while here we treat the topic per se, in an accompanying paper [3] we use (an extended version of) the theory to prove a strengthening of a tiling theorem of Komlós [5]. We expect further similar applications of our graphon approach to extremal graph theory in the future.…”