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.
@article{10_37236_6241,
author = {Martin Dole\v{z}al and Jan Hladk\'y},
title = {Matching polytons},
journal = {The electronic journal of combinatorics},
year = {2019},
volume = {26},
number = {4},
doi = {10.37236/6241},
zbl = {1428.05243},
url = {http://geodesic.mathdoc.fr/articles/10.37236/6241/}
}
TY - JOUR
AU - Martin Doležal
AU - Jan Hladký
TI - Matching polytons
JO - The electronic journal of combinatorics
PY - 2019
VL - 26
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/6241/
DO - 10.37236/6241
ID - 10_37236_6241
ER -
%0 Journal Article
%A Martin Doležal
%A Jan Hladký
%T Matching polytons
%J The electronic journal of combinatorics
%D 2019
%V 26
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/6241/
%R 10.37236/6241
%F 10_37236_6241
Martin Doležal; Jan Hladký. Matching polytons. The electronic journal of combinatorics, Tome 26 (2019) no. 4. doi: 10.37236/6241
