Tilings in graphons

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].

[1]  Brett Stevens,et al.  Digraphs are 2-Weight Choosable , 2011, Electron. J. Comb..

[2]  János Komlós,et al.  Tiling Turán Theorems , 2000, Comb..

[3]  Gabor Lippner,et al.  Borel oracles. An analytical approach to constant-time algorithms , 2009, 0907.1805.

[4]  B. Szegedy,et al.  Testing properties of graphs and functions , 2008, 0803.1248.

[5]  Independent sets, cliques, and colorings in graphons , 2017, Eur. J. Comb..

[6]  Peter Allen,et al.  A Density Corrádi–Hajnal Theorem , 2011, Canadian Journal of Mathematics.

[7]  Kellen Petersen August Real Analysis , 2009 .

[8]  Béla Bollobás,et al.  The independence ratio of regular graphs , 1981 .

[9]  Alexander A. Razborov,et al.  On the Minimal Density of Triangles in Graphs , 2008, Combinatorics, Probability and Computing.

[10]  Yoshiharu Kohayakawa,et al.  Small subsets inherit sparse ε-regularity , 2004 .

[11]  László Lovász,et al.  Limits of dense graph sequences , 2004, J. Comb. Theory B.

[12]  Jan Hladký,et al.  Komlós's tiling theorem via graphon covers , 2016, J. Graph Theory.

[13]  P. Erdos,et al.  On maximal paths and circuits of graphs , 1959 .

[14]  Stephen A. Clark An Infinite-Dimensional LP Duality Theorem , 2003, Math. Oper. Res..

[15]  Jonathan R. Partington An Epsilon of Room, I: Real Analysis (pages from year three of a mathematical blog) (Graduate Studies in Mathematics 117) , 2012 .

[16]  M. Bálek,et al.  Large Networks and Graph Limits , 2022 .

[17]  J. Hladký,et al.  Matching Polytons , 2016, Electron. J. Comb..

[18]  Marc Lelarge,et al.  Matchings on infinite graphs , 2011, 1102.0712.

[19]  V. Sós,et al.  Convergent Sequences of Dense Graphs I: Subgraph Frequencies, Metric Properties and Testing , 2007, math/0702004.

[20]  Alexander A. Razborov,et al.  Flag algebras , 2007, Journal of Symbolic Logic.

[21]  Jan Hladký,et al.  First steps in combinatorial optimization on graphons: Matchings , 2017, Electron. Notes Discret. Math..

[22]  László Lovász Subgraph Densities in Signed Graphons and the Local Simonovits-Sidorenko Conjecture , 2011, Electron. J. Comb..

[23]  Krzysztof Onak,et al.  Constant-Time Approximation Algorithms via Local Improvements , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[24]  Yoshiharu Kohayakawa,et al.  Small subsets inherit sparse epsilon-regularity , 2007, J. Comb. Theory, Ser. B.

[25]  S. Y. Wu,et al.  Extremal points and optimal solutions for general capacity problems , 1992, Math. Program..