205 research outputs found

    The extremal function for partial bipartite tilings

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    For a fixed bipartite graph H and given number c, 0<c<1, we determine the threshold T_H(c) which guarantees that any n-vertex graph with at edge density at least T_H(c) contains (1o(1))c/v(H)n(1-o(1))c/v(H) n vertex-disjoint copies of H. In the proof we use a variant of a technique developed by Komlos~\bcolor{[Combinatorica 20 (2000), 203-218}]Comment: 10 page

    A density Corr\'adi-Hajnal Theorem

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    We find, for all sufficiently large nn and each kk, the maximum number of edges in an nn-vertex graph which does not contain k+1k+1 vertex-disjoint triangles. This extends a result of Moon [Canad. J. Math. 20 (1968), 96-102] which is in turn an extension of Mantel's Theorem. Our result can also be viewed as a density version of the Corradi-Hajnal Theorem.Comment: 41 pages (including 11 pages of appendix), 4 figures, 2 table

    On the Existence of Loose Cycle Tilings and Rainbow Cycles

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    abstract: Extremal graph theory results often provide minimum degree conditions which guarantee a copy of one graph exists within another. A perfect FF-tiling of a graph GG is a collection F\mathcal{F} of subgraphs of GG such that every element of F\mathcal{F} is isomorphic to FF and such that every vertex in GG is in exactly one element of F\mathcal{F}. Let Ct3C^{3}_{t} denote the loose cycle on t=2st = 2s vertices, the 33-uniform hypergraph obtained by replacing the edges e={u,v}e = \{u, v\} of a graph cycle CC on ss vertices with edge triples {u,xe,v}\{u, x_e, v\}, where xex_e is uniquely assigned to ee. This dissertation proves for even t6t \geq 6, that any sufficiently large 33-uniform hypergraph HH on ntZn \in t \mathbb{Z} vertices with minimum 11-degree \delta^1(H) \geq {n - 1 \choose 2} - {\Bsize \choose 2} + c(t,n) + 1, where c(t,n){0,1,3}c(t,n) \in \{0, 1, 3\}, contains a perfect Ct3C^{3}_{t}-tiling. The result is tight, generalizing previous results on C43C^3_4 by Han and Zhao. For an edge colored graph GG, let the minimum color degree δc(G)\delta^c(G) be the minimum number of distinctly colored edges incident to a vertex. Call GG rainbow if every edge has a unique color. For 5\ell \geq 5, this dissertation proves that any sufficiently large edge colored graph GG on nn vertices with δc(G)n+12\delta^c(G) \geq \frac{n + 1}{2} contains a rainbow cycle on \ell vertices. The result is tight for odd \ell and extends previous results for =3\ell = 3. In addition, for even 4\ell \geq 4, this dissertation proves that any sufficiently large edge colored graph GG on nn vertices with δc(G)n+c()3\delta^c(G) \geq \frac{n + c(\ell)}{3}, where c(){5,7}c(\ell) \in \{5, 7\}, contains a rainbow cycle on \ell vertices. The result is tight when 66 \nmid \ell. As a related result, this dissertation proves for all 4\ell \geq 4, that any sufficiently large oriented graph DD on nn vertices with δ+(D)n+13\delta^+(D) \geq \frac{n + 1}{3} contains a directed cycle on \ell vertices. This partially generalizes a result by Kelly, K\"uhn, and Osthus that uses minimum semidegree rather than minimum out degree.Dissertation/ThesisDoctoral Dissertation Mathematics 201

    How quickly can we sample a uniform domino tiling of the 2L x 2L square via Glauber dynamics?

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    TThe prototypical problem we study here is the following. Given a 2L×2L2L\times 2L square, there are approximately exp(4KL2/π)\exp(4KL^2/\pi ) ways to tile it with dominos, i.e. with horizontal or vertical 2×12\times 1 rectangles, where K0.916K\approx 0.916 is Catalan's constant [Kasteleyn '61, Temperley-Fisher '61]. A conceptually simple (even if computationally not the most efficient) way of sampling uniformly one among so many tilings is to introduce a Markov Chain algorithm (Glauber dynamics) where, with rate 11, two adjacent horizontal dominos are flipped to vertical dominos, or vice-versa. The unique invariant measure is the uniform one and a classical question [Wilson 2004,Luby-Randall-Sinclair 2001] is to estimate the time TmixT_{mix} it takes to approach equilibrium (i.e. the running time of the algorithm). In [Luby-Randall-Sinclair 2001, Randall-Tetali 2000], fast mixin was proven: Tmix=O(LC)T_{mix}=O(L^C) for some finite CC. Here, we go much beyond and show that cL2TmixL2+o(1)c L^2\le T_{mix}\le L^{2+o(1)}. Our result applies to rather general domain shapes (not just the 2L×2L2L\times 2L square), provided that the typical height function associated to the tiling is macroscopically planar in the large LL limit, under the uniform measure (this is the case for instance for the Temperley-type boundary conditions considered in [Kenyon 2000]). Also, our method extends to some other types of tilings of the plane, for instance the tilings associated to dimer coverings of the hexagon or square-hexagon lattices.Comment: to appear on PTRF; 42 pages, 9 figures; v2: typos corrected, references adde

    A (2+1)-dimensional growth process with explicit stationary measures

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    We introduce a class of (2+1)-dimensional stochastic growth processes, that can be seen as irreversible random dynamics of discrete interfaces. "Irreversible" means that the interface has an average non-zero drift. Interface configurations correspond to height functions of dimer coverings of the infinite hexagonal or square lattice. The model can also be viewed as an interacting driven particle system and in the totally asymmetric case the dynamics corresponds to an infinite collection of mutually interacting Hammersley processes. When the dynamical asymmetry parameter (pq)(p-q) equals zero, the infinite-volume Gibbs measures πρ\pi_\rho (with given slope ρ\rho) are stationary and reversible. When pqp\ne q, πρ\pi_\rho are not reversible any more but, remarkably, they are still stationary. In such stationary states, we find that the average height function at any given point xx grows linearly with time tt with a non-zero speed: EQx(t):=E(hx(t)hx(0))=V(ρ)t\mathbb E Q_x(t):=\mathbb E(h_x(t)-h_x(0))= V(\rho) t while the typical fluctuations of Qx(t)Q_x(t) are smaller than any power of tt as tt\to\infty. In the totally asymmetric case of p=0,q=1p=0,q=1 and on the hexagonal lattice, the dynamics coincides with the "anisotropic KPZ growth model" introduced by A. Borodin and P. L. Ferrari. For a suitably chosen, "integrable", initial condition (that is very far from the stationary state), they were able to determine the hydrodynamic limit and a CLT for interface fluctuations on scale logt\sqrt{\log t}, exploiting the fact that in that case certain space-time height correlations can be computed exactly.Comment: 37 pages, 13 figures. v3: some references added, introduction expanded, minor changes in the bul

    Matchings in 3-uniform hypergraphs

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    We determine the minimum vertex degree that ensures a perfect matching in a 3-uniform hypergraph. More precisely, suppose that H is a sufficiently large 3-uniform hypergraph whose order n is divisible by 3. If the minimum vertex degree of H is greater than \binom{n-1}{2}-\binom{2n/3}{2}, then H contains a perfect matching. This bound is tight and answers a question of Han, Person and Schacht. More generally, we show that H contains a matching of size d\le n/3 if its minimum vertex degree is greater than \binom{n-1}{2}-\binom{n-d}{2}, which is also best possible. This extends a result of Bollobas, Daykin and Erdos.Comment: 18 pages, 1 figure. To appear in JCT
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