Matching Polytons

Abstract

Hladký, Hu, and Piguet [Tilings in graphons, preprint] introduced the notions of matching and fractional vertex covers in graphons. These are counterparts to the corresponding notions in finite graphs.Combinatorial optimization studies the structure of the matching polytope and the fractional vertex cover polytope of a graph. Here, in analogy, we initiate the study of the structure of the set of all matchings and of all fractional vertex covers in a graphon. We call these sets the matching polyton and the fractional vertex cover polyton.We also study properties of matching polytons and fractional vertex cover polytons along convergent sequences of graphons.As an auxiliary tool of independent interest, we prove that a graphon is r-partite if and only if it contains no graph of chromatic number r + 1. This in turn gives a characterization of bipartite graphons as those having a symmetric spectrum. 1 1 the matching ratio is just the matching number divided by the number of vertices 2 See Section 2 for the definition of cut-distance convergence. 3 These applications become particularly interesting when general F -tilings are considered.

Komlós's tiling theorem via graphon covers

“…The original proof of Theorem 1.2 is not lengthy but uses an ingenious recursive regularization of the graph . 4 Our proof offers an alternative point of view on the problem. In fact we believe it follows the most natural strategy: If had only a small tiling number then, by the LP duality, 5 it would have a small fractional -cover.…”

“…Now assume that for some > 0 there is a set of measure at least such that ( ) ⊆ ( 1 + , 1]. Fix = ( (1− ) 4ℎ ) 4 . Then…”

“…Let us remark that in , the authors provide a graphon proof of the Erdős–Gallai Theorem. The key tool to this end is to establish the half‐integrality property of the fractional vertex cover “polyt on .” These objects are defined in analogy to fractional vertex cover potypes of graphs, but for graphons (hence the “‐ on ” ending).…”

Independent sets, cliques, and colorings in graphons

“…(b) For the sequence (G n ∼ G(n, W)) we have χ(W) = lim n→∞ χ(G n ), almost surely. The next proposition, which can be found in [6,Proposition 5], is the key towards proving Theorem 14. We remark that the proof of this proposition given in [6] is short but non-trivial.…”

“…We can now take the closure (in the weak* topology) of the convex hull of such functions and get what we call independent set polyton IND(W) ⊂ R Ω . Such a graphon approach to polyhedral combinatorics has been introduced [6], namely for the so-called matching polytope/polyton. To illustrate the potential of this area, let us prove a part of a counterpart to Proposition 8.…”