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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: Komloss tiling theorem via graphon covers, preprint] we give a proof of a strengthening of a theorem of Komlos [Komlos: Tiling Turan Theorems, Combinatorica, 2000].
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We determine the asymptotic behaviour of the chromatic number of exchangeable random graphs defined by step-regulated graphons. Furthermore, we show that the upper bound holds for a general graphon. We also extend these results to sparse random graphs obtained by percolations on graphons.
A combinatorial substitution is a map over tilings which allows to define sets of tilings with a strong hierarchical structure. In this paper, we show that such sets of tilings are sofic, that is, can be enforced by finitely many local constraints. This extends some similar previous results (Mozes90, Goodman-Strauss98) in a much shorter presentation.
For a $k$-vertex graph $F$ and an $n$-vertex graph $G$, an $F$-tiling in $G$ is a collection of vertex-disjoint copies of $F$ in $G$. For $rin mathbb{N}$, the $r$-independence number of $G$, denoted $alpha_r(G)$ is the largest size of a $K_r$-free set of vertices in $G$. In this paper, we discuss Ramsey--Turan-type theorems for tilings where one is interested in minimum degree and independence number conditions (and the interaction between the two) that guarantee the existence of optimal $F$-tilings. For cliques, we show that for any $kgeq 3$ and $eta>0$, any graph $G$ on $n$ vertices with $delta(G)geq eta n$ and $alpha_k(G)=o(n)$ has a $K_k$-tiling covering all but $lfloortfrac{1}{eta}rfloor(k-1)$ vertices. All conditions in this result are tight; the number of vertices left uncovered can not be improved and for $eta<tfrac{1}{k}$, a condition of $alpha_{k-1}(G)=o(n)$ would not suffice. When $eta>tfrac{1}{k}$, we then show that $alpha_{k-1}(G)=o(n)$ does suffice, but not $alpha_{k-2}(G)=o(n)$. These results unify and generalise previous results of Balogh-Molla-Sharifzadeh, Nenadov-Pehova and Balogh-McDowell-Molla-Mycroft on the subject. We further explore the picture when $F$ is a tree or a cycle and discuss the effect of replacing the independence number condition with $alpha^*(G)=o(n)$ (meaning that any pair of disjoint linear sized sets induce an edge between them) where one can force perfect $F$-tilings covering all the vertices. Finally we discuss the consequences of these results in the randomly perturbed setting.
A semi-regular tiling of the hyperbolic plane is a tessellation by regular geodesic polygons with the property that each vertex has the same vertex-type, which is a cyclic tuple of integers that determine the number of sides of the polygons surrounding the vertex. We determine combinatorial criteria for the existence, and uniqueness, of a semi-regular tiling with a given vertex-type, and pose some open questions.
Let $vec{T}_k$ be the transitive tournament on $k$ vertices. We show that every oriented graph on $n=4m$ vertices with minimum total degree $(11/12+o(1))n$ can be partitioned into vertex disjoint $vec{T}_4$s, and this bound is asymptotically tight. We also improve the best known bound on the minimum total degree for partitioning oriented graphs into vertex disjoint $vec{T}_k$s.
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