Tiling in bipartite graphs with asymmetric minimum degrees
Andrzej Czygrinow, Louis DeBiasio
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34 pages, 4 figures. This is the unabridged version of the paper, containing the full proof of Theorem 1.7. The case when |δ_U-δ_V| is small and s>2 involves a lengthy case analysis, spanning pages 20-32; this section is not included in the "journal version"
The problem of determining the optimal minimum degree condition for a balanced bipartite graph on 2ms vertices to contain m vertex disjoint copies of K_{s,s} was solved by Zhao. Later Hladký and Schacht, and Czygrinow and DeBiasio determined the optimal minimum degree condition for a balanced bipartite graph on 2m(s+t) vertices to contain m vertex disjoint copies of K_{s,t} for fixed positive integers s<t.
For a balanced bipartite graph G[U,V], let δ_U be the minimum degree over all vertices in U and δ_V be the minimum degree over all vertices in V. We consider the problem of determining the optimal value of δ_U+δ_V which guarantees that G can be tiled with K_{s,s}. We show that the optimal value depends on D:=|δ_V-δ_U|. When D is small, we show that δ_U+δ_V\geq n+3s-5 is best possible. As D becomes larger, we show that δ_U+δ_V can be made smaller, but no smaller than n+2s-2s^{1/2}. However, when D=n-C for some constant C, we show that there exist graphs with δ_U+δ_V\geq n+s^{s^{1/3}} which cannot be tiled with K_{s,s}.
For any bipartite graph $H$, we determine a minimum degree threshold for a balanced bipartite graph $G$ to contain a perfect $H$-tiling. We show that this threshold is best possible up to a constant depending only on $H$. Additionally, we prove a corresponding minimum degree threshold to guarantee that $G$ has an $H$-tiling missing only a constant number of vertices. Our threshold for the perfect tiling depends on either the chromatic number $χ(H)$ or the critical chromatic number $χ_{cr}(H)$ while the threshold for the almost perfect tiling only depends on $χ_{cr}(H)$. Our results answer two …
Bipartite graph tiling was studied by Zhao who gave the best possible minimum degree conditions for a balanced bipartite graph on 2ms vertices to contain m vertex disjoint copies of K_{s,s}. Let s<t be fixed positive integers. Hladký and Schacht gave minimum degree conditions for a balanced bipartite graph on 2m(s+t) vertices to contain m vertex disjoint copies of K_{s,t}. Their results were best possible, except in the case when m is odd and t> 2s+1. We give the best possible minimum degree condition in this case.
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 $(1-o(1))c/v(H) n$ vertex-disjoint copies of H. In the proof we use a variant of a technique developed by Komlos~\bcolor{[Combinatorica 20 (2000), 203-218}]
Let s<t be two fixed positive integers. We study what are the minimum degree conditions for a bipartite graph G, with both color classes of size n=k(s+t), which ensure that G has a K_{s,t}-factor. Exact result for large n is given. Our result extends the work of Zhao, who determined the minimum degree threshold which guarantees that a bipartite graph has a K_{s,s}-factor.
We prove that for any $r\in \mathbb{N}$, there exists a constant $C_r$ such that the following is true. Let $\mathcal{F}=\{F_1,F_2,\dots\}$ be an infinite sequence of bipartite graphs such that $|V(F_i)|=i$ and $Δ(F_i)\leq Δ$ hold for all $i$. Then in any $r$-edge coloured complete graph $K_n$, there is a collection of at most $\exp(C_rΔ)$ monochromatic subgraphs, each of which is isomorphic to an element of $\mathcal{F}$, whose vertex sets partition $V(K_n)$. This proves a conjecture of Corsten and Mendonça in a strong form and generalizes results on the multicolour Ramsey numbers of bounded-…
A technique called graphical condensation is used to prove various combinatorial identities among numbers of (perfect) matchings of planar bipartite graphs and tilings of regions. Graphical condensation involves superimposing matchings of a graph onto matchings of a smaller subgraph, and then re-partitioning the united matching (actually a multigraph) into matchings of two other subgraphs, in one of two possible ways. This technique can be used to enumerate perfect matchings of a wide variety of bipartite planar graphs. Applications include domino tilings of Aztec diamonds and rectangles, diab…
Generalizing results of Temperley, Brooks, Smith, Stone and Tutte and others we describe a natural equivalence between three planar objects: weighted bipartite planar graphs; planar Markov chains; and tilings with convex polygons. This equivalence provides a measure-preserving bijection between dimer coverings of a weighted bipartite planar graph and spanning trees on the corresponding Markov chain. The tilings correspond to harmonic functions on the Markov chain and to ``discrete analytic functions'' on the bipartite graph. The equivalence is extended to infinite periodic graphs, and we class…
We study whether an asymmetric limited-magnitude ball may tile $\mathbb{Z}^n$. This ball generalizes previously studied shapes: crosses, semi-crosses, and quasi-crosses. Such tilings act as perfect error-correcting codes in a channel which changes a transmitted integer vector in a bounded number of entries by limited-magnitude errors. A construction of lattice tilings based on perfect codes in the Hamming metric is given. Several non-existence results are proved, both for general tilings, and lattice tilings. A complete classification of lattice tilings for two certain cases is proved.
An $n$-dimensional chair consists of an $n$-dimensional box from which a smaller $n$-dimensional box is removed. A tiling of an $n$-dimensional chair has two nice applications in coding for write-once memories. The first one is in the design of codes which correct asymmetric errors with limited-magnitude. The second one is in the design of $n$ cells $q$-ary write-once memory codes. We show an equivalence between the design of a tiling with an integer lattice and the design of a tiling from a generalization of splitting (or of Sidon sequences). A tiling of an $n$-dimensional chair can define a …
F-nomial coefficients encompass among others well-known binomial coefficients or Gaussian coefficients that count subsets of finite set and subspaces of finite vector space respectively. Here, the so called F-cobweb tiling sequences N(a) are considered. For such specific sequences a new interpretation with respect to Kwasniewski general combinatorial interpretation of F-nomial coefficients is unearhed. Namely, for tiling sequences F = N(a)$ the F-nomial coefficients are equal to the number of labeled special bipartite multigraphs denoted here as a-multigraphs G(a,n,k). An explicit relation bet…
We introduce a class of graphs called compound graphs, generalizing rectangles, which are constructed out of copies of a planar bipartite base graph. The main result is that the number of perfect matchings of every compound graph is divisible by the number of matchings of its base graph. Our approach is to use Kasteleyn's theorem to prove a key lemma, from which the divisibility theorem follows combinatorially. This theorem is then applied to provide a proof of Problem 21 of Propp's Enumeration of Matchings, a divisibility property of rectangles. Finally, we present a new proof, in the same sp…
Let $k,l,m$ be integers and $r(k,l,m)$ be the minimum integer $N$ such that for any red-blue-green coloring of $K_{N,N}$, there is a red matching of size at least $k$ in a component, or a blue matching of at least size $l$ in a component, or a green matching of size at least $m$ in a component. In this paper, we determine the exact value of $r(k,l,m)$ completely. Applying a technique originated by Łuczak that applies Szemerédi's Regularity Lemma to reduce the problem of showing the existence of a monochromatic cycle to show the existence of a monochromatic matching in a component, we obtain th…
Ehrenborg noted that all tilings of a bipartite planar graph are encoded by its cubical matching complex and claimed that this complex is collapsible. We point out to an oversight in his proof and explain why these complexes can be the union of collapsible complexes. Also, we prove that all links in these complexes are suspensions up to homotopy. Furthermore, we extend the definition of a cubical matching complex to planar graphs that are not necessarily bipartite, and show that these complexes are either contractible or a disjoint union of contractible complexes. For a simple connected region…
Solving the kernel perfect problem by (simple) forbidden subdigraphs for digraphs in some families of generalized tournaments and generalized bipartite tournaments
We consider the dimer problem on a non-bipartite graph $G$, where there are two types of dimers one of which we regard impurities. Results of simulations using Markov chain seem to indicate that impurities are tend to distribute on the boundary, which we set as a conjecture. We first show that there is a bijection between the set of dimer coverings on $G$ and the set of spanning forests on two graphs which are made from $G$, with configuration of impurities satisfying a pairing condition. This bijection can be regarded as a extension of the Temperley bijection. We consider local move consistin…
In this paper we consider the separability problem for bipartite quantum states arising from graphs. Earlier it was proved that the degree criterion is the graph-theoretic counterpart of the familiar positive partial transpose criterion for separability, although there are entangled states with positive partial transpose for which the degree criterion fails. Here we introduce the concept of partially symmetric graphs and degree symmetric graphs by using the well-known concept of partial transposition of a graph and degree criteria, respectively. Thus, we provide classes of bipartite separable …
A proper edge $t$-coloring of a graph $G$ is a coloring of edges of $G$ with colors $1,2,...,t$ such that all colors are used, and no two adjacent edges receive the same color. The set of colors of edges incident with a vertex $x$ is called a spectrum of $x$. An arbitrary nonempty subset of consecutive integers is called an interval. Suppose that all edges of a graph $G$ are colored in the game of Alice and Bob with asymmetric distribution of roles. Alice determines the number $t$ of colors in the future proper edge coloring of $G$ and aspires to minimize the number of vertices with an interva…
The method of hypergraph containers, introduced recently by Balogh, Morris, and Samotij, and independently by Saxton and Thomason, has proved to be an extremely useful tool in the study of various monotone graph properties. In particular, a fairly straightforward application of this technique allows one to locate, for each non-bipartite graph $H$, the threshold at which the distribution of edges in a typical $H$-free graph with a given number of edges undergoes a transition from 'random-like' to 'structured'. On the other hand, for non-monotone hereditary graph properties the standard version …
We explore the limiting empirical eigenvalue distributions arising from matrices of the form \[A_{n+1} = \begin{bmatrix} A_n & I\\ I & A_n \end{bmatrix} , \]where $A_0$ is the adjacency matrix of a $k$-regular graph. We find that for bipartite graphs, the distributions are centered symmetric binomial distributions, and for non-bipartite graphs, the distributions are asymmetric. This research grew out of our work on neural networks in $k$-regular graphs. Our original question was whether or not the graph cylinder construction would produce an expander graph that is a suitable candidate for the …
For two graphs $X$ and $Y$ with vertex sets $V(X)$ and $V(Y)$ of the same cardinality $n,$ the friends-and-strangers graph $\mathsf{FS}(X,Y)$ was recently defined by Defant and Kravitz. The vertices of $\mathsf{FS}(X,Y)$ are the bijections from $V(X)$ to $V(Y),$ and two bijections $σ$ and $τ$ are adjacent if they agree everywhere except at two vertices $a,b\in V(X)$ such that $a$ and $b$ are adjacent in $X$ and $σ(a)$ and $σ(b)$ are adjacent in $Y.$ We study generalized versions of two problems by Alon, Defant, and Kravitz. First, we show that if $X$ and $Y$ have minimum degrees $δ(X)$ and $δ(…
