The bandwidth of a graph G is the minimum of the maximum difference between adjacent labels when the vertices have distinct integer labels. We provide a polynomial algorithm to produce an optimal bandwidth labeling for graphs in a special class of block graphs (graphs in which every block is a clique), namely those where deleting the vertices of degree one produces a path of cliques. The result is best possible in various ways. Furthermore, for two classes of graphs that are ``almost'' caterpillars, the bandwidth problem is NP-complete.
We develop a tool for embedding almost spanning degenerate graphs of small bandwidth. As an application, we extend the blow-up lemma to degenerate graphs of small bandwidth, the bandwidth theorem to degenerate graphs, and make progress on a conjecture of Burr and Erdős on Ramsey number of degenerate graphs.
The edge-bandwidth of a graph is the minimum, over all labelings of the edges with distinct integers, of the maximum difference between labels of two incident edges. We prove that edge-bandwidth is at least as large as bandwidth for every graph, with equality for certain caterpillars. We obtain sharp or nearly-sharp bounds on the change in edge-bandwidth under addition, subdivision, or contraction of edges. We compute edge-bandwidth for cliques, bicliques, caterpillars, and some theta graphs.
We estimate Ramsey numbers for bipartite graphs with small bandwidth and bounded maximum degree. In particular we determine asymptotically the two and three color Ramsey numbers for grid graphs. More generally, we determine asymptotically the two color Ramsey number for bipartite graphs with small bandwidth and bounded maximum degree and the three color Ramsey number for such graphs with the additional assumption that the bipartite graph is balanced.
The Bandwidth theorem of Böttcher, Schacht and Taraz gives a condition on the minimum degree of an $n$-vertex graph $G$ that ensures $G$ contains every $r$-chromatic graph $H$ on $n$ vertices of bounded degree and of bandwidth $o(n)$, thereby proving a conjecture of Bollobás and Komlós. In this paper we prove a version of the Bandwidth theorem for locally dense graphs. Indeed, we prove that every locally dense $n$-vertex graph $G$ with $δ(G) > (1/2+o(1))n$ contains as a subgraph any given (spanning) $H$ with bounded maximum degree and sublinear bandwidth.
Institut für Informatik, Universität Heidelberg（海德堡大学）；Faculty of Computer Science and Mathematics, University of Passau（帕绍大学）；Department of Mathematics, London School of Economics（伦敦政治经济学院）；Zuse Institute Berlin（柏林祖斯研究所）；Department of Mathematics, University College London（伦敦大学学院）
We estimate the $3$-colour bipartite Ramsey number for balanced bipartite graphs $H$ with small bandwidth and bounded maximum degree. More precisely, we show that the minimum value of $N$ such that in any $3$-edge colouring of $K_{N,N}$ there is a monochromatic copy of $H$ is at most $\big(3/2+o(1)\big)|V(H)|$. In particular, we determine asymptotically the $3$-colour bipartite Ramsey number for balanced grid graphs.
The Bandwidth Theorem of Böttcher, Schacht and Taraz [Mathematische Annalen 343 (1), 175-205] gives minimum degree conditions for the containment of spanning graphs H with small bandwidth and bounded maximum degree. We generalise this result to a-arrangeable graphs H with Δ(H)<sqrt(n)/log(n), where n is the number of vertices of H. Our result implies that sufficiently large n-vertex graphs G with minimum degree at least (3/4+γ)n contain almost all planar graphs on n vertices as subgraphs. Using techniques developed by Allen, Brightwell and Skokan [Combinatorica, to appear] we can also apply ou…
In this paper we construct a class of bounded degree bipartite graphs with a small separator and large bandwidth. Furthermore, we also prove that graphs from this class are spanning subgraphs of graphs with minimum degree just slightly larger than $n/2$.
We establish relations between the bandwidth and the treewidth of bounded degree graphs G, and relate these parameters to the size of a separator of G as well as the size of an expanding subgraph of G. Our results imply that if one of these parameters is sublinear in the number of vertices of G then so are all the others. This implies for example that graphs of fixed genus have sublinear bandwidth or, more generally, a corresponding result for graphs with any fixed forbidden minor. As a consequence we establish a simple criterion for universality for such classes of graphs and show for example…
Let $n,k,b$ be integers with $1 \le k-1 \le b \le n$ and let $G_{n,k,b}$ be the graph whose vertices are the $k$-element subsets $X$ of $\{0,\dots,n\}$ with $\max(X)-\min(X) \le b$ and where two such vertices $X,Y$ are joined by an edge if $\max(X \cup Y) - \min(X \cup Y) \le b$. These graphs are generated by applying a transformation to maximal $k$-uniform hypergraphs of bandwidth $b$ that is used to reduce the (weak) edge clique covering problem to a vertex clique covering problem. The bandwidth of $G_{n,k,b}$ is thus the largest possible bandwidth of any transformed $k$-uniform hypergraph o…
Boettcher, Schacht and Taraz gave a condition on the minimum degree of a graph G on n vertices that ensures G contains every r-chromatic graph H on n vertices of bounded degree and of bandwidth o(n), thereby proving a conjecture of Bollobas and Komlos. We strengthen this result in the case when H is bipartite. Indeed, we give an essentially best-possible condition on the degree sequence of a graph G on n vertices that forces G to contain every bipartite graph H on n vertices of bounded degree and of bandwidth o(n). This also implies an Ore-type result. In fact, we prove a much stronger result …
The bandwidth theorem [Mathematische Annalen, 343(1):175--205, 2009] states that any $n$-vertex graph $G$ with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)n$ contains all $n$-vertex $k$-colourable graphs $H$ with bounded maximum degree and bandwidth $o(n)$. We provide sparse analogues of this statement in random graphs as well as pseudorandom graphs. More precisely, we show that for $p\gg \big(\tfrac{\log n}{n}\big)^{1/Δ}$ asymptotically almost surely each spanning subgraph $G$ of $G(n,p)$ with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)pn$ contains all $n$-vertex $k$-colourable graphs $H$ …
The bandwidth theorem [Mathematische Annalen, 343(1):175--205, 2009] states that any $n$-vertex graph $G$ with minimum degree $(\frac{k-1}{k}+o(1))n$ contains all $n$-vertex $k$-colourable graphs $H$ with bounded maximum degree and bandwidth $o(n)$. In [arXiv:1612.00661] a random graph analogue of this statement is proved: for $p\gg (\frac{\log n}{n})^{1/Δ}$ a.a.s. each spanning subgraph $G$ of $G(n,p)$ with minimum degree $(\frac{k-1}{k}+o(1))pn$ contains all $n$-vertex $k$-colourable graphs $H$ with maximum degree $Δ$, bandwidth $o(n)$, and at least $C p^{-2}$ vertices not contained in any t…
Given a collection $\mathcal{G}=(G_1,\dots, G_h)$ of graphs on the same vertex set $V$ of size $n$, an $h$-edge graph $H$ on the vertex set $V$ is a $\mathcal{G}$-transversal if there exists a bijection $λ: E(H) \rightarrow [h]$ such that $e\in E(G_{λ(e)})$ for each $e\in E(H)$. The conditions on the minimum degree $δ(\mathcal{G})=\min_{i\in[h]}\{ δ(G_i)\}$ for finding a spanning $\mathcal{G}$-transversal isomorphic to a graph $H$ have been actively studied when $H$ is a Hamilton cycle, an $F$-factor, a spanning tree with maximum degree $o(n/\log n)$ and a power of a Hamilton cycle, etc. In th…
A graph $G$ is said to have \textit{bandwidth} at most $b$, if there exists a labeling of the vertices by $1,2,..., n$, so that $|i - j| \leq b$ whenever $\{i,j\}$ is an edge of $G$. Recently, Böttcher, Schacht, and Taraz verified a conjecture of Bollobás and Komlós which says that for every positive $r,Δ,γ$, there exists $β$ such that if $H$ is an $n$-vertex $r$-chromatic graph with maximum degree at most $Δ$ which has bandwidth at most $βn$, then any graph $G$ on $n$ vertices with minimum degree at least $(1 - 1/r + γ)n$ contains a copy of $H$ for large enough $n$. In this paper, we extend t…
Let $G = (V,E)$ be a graph on $n$ vertices and $f: V\rightarrow [1,n]$ a one to one map of $V$ onto the integers $1$ through $n$. Let $dilation(f) =$ max$\{ |f(v) - f(w)|: vw\in E \}$. Define the {\it bandwidth} $B(G)$ of $G$ to be the minimum possible value of $dilation(f)$ over all such one to one maps $f$. Next define the {\it Kneser Graph} $K(n,r)$ to be the graph with vertex set $\binom{[n]}{r}$, the collection of $r$-subsets of an $n$ element set, and edge set $E = \{ vw: v,w\in \binom{[n]}{r}, v\cap w = \emptyset \}$. For fixed $r\geq 4$ and $n\rightarrow \infty$ we show that $$B(K(n,r)…
A graph $\mathcal{H}=(W,E_\mathcal{H})$ is said to have {\em bandwidth} at most $b$ if there exists a labeling of $W$ as $w_1,w_2,\dots,w_n$ such that $|i-j|\leq b$ for every edge $w_iw_j\in E_\mathcal{H}$. We say that $\mathcal{H}$ is a {\em balanced $(β,Δ)$-graph} if it is a bipartite graph with bandwidth at most $β|W|$ and maximum degree at most $Δ$, and it also has a proper 2-coloring $χ:W\rightarrow[2]$ such that $||χ^{-1}(1)|-|χ^{-1}(2)||\leqβ|χ^{-1}(2)|$. In this paper, we prove that for every $γ>0$ and every natural number $Δ$, there exists a constant $β>0$ such that for every balanced…
We provide a degree condition on a regular $n$-vertex graph $G$ which ensures the existence of a near optimal packing of any family $\mathcal H$ of bounded degree $n$-vertex $k$-chromatic separable graphs into $G$. In general, this degree condition is best possible. Here a graph is separable if it has a sublinear separator whose removal results in a set of components of sublinear size. Equivalently, the separability condition can be replaced by that of having small bandwidth. Thus our result can be viewed as a version of the bandwidth theorem of Böttcher, Schacht and Taraz in the setting of ap…
A connected graph is called a multi-block graph if each of its blocks is a complete multi-partite graph. Building on the work of \cite{Bp3,Hou3}, we compute the determinant and inverse of the distance matrix for a class of multi-block graphs.
We derive a new lower bound for the bandwidth of a graph that is based on a new lower bound for the minimum cut problem. Our new semidefinite programming relaxation of the minimum cut problem is obtained by strengthening the known semidefinite programming relaxation for the quadratic assignment problem (or for the graph partition problem) by fixing two vertices in the graph; one on each side of the cut. This fixing results in several smaller subproblems that need to be solved to obtain the new bound. In order to efficiently solve these subproblems we exploit symmetry in the data; that is, both…
A class of graphs is called block-stable when a graph is in the class if and only if each of its blocks is. We show that, as for trees, for most $n$-vertex graphs in such a class, each vertex is in at most $(1+o(1)) \log n / \log\log n$ blocks, and each path passes through at most $5 (n \log n)^{1/2}$ blocks. These results extend to `weakly block-stable' classes of graphs.
In this paper, we provide a natural bijection between a special family of block circulant graphs and the graphs of critical pairs of the posets known as generalized crowns. In particular, every graph in this family of block circulant graphs we investigate has a generating block row that follows a symmetric growth pattern of the all ones matrix. The natural bijection provides an upper bound on the chromatic number for this infinite family of graphs.
A connected graph is called a bi-block graph if each of its blocks is a complete bipartite graph. Let $\mathcal{B}(\mathbf{k}, α)$ be the class of bi-block graph on $\mathbf{k}$ vertices with given independence number $α$. It is easy to see that every bi-block graph is a bipartite graph. For a bipartite graph $G$ on $\mathbf{k}$ vertices, the independence number $α(G)$ satisfies $\ceil*{\frac{\mathbf{k}}{2}} \leq α(G) \leq \mathbf{k}-1$. In this article, we prove that the maximum spectral radius $ρ(G)$ among all graphs $G$ in $\mathcal{B}(\mathbf{k}, α)$, is uniquely attained for the complete …
Using the theory of combinatorial species, we compute the cycle index for bipartite graphs, which we use to count unlabeled bipartite graphs and bipartite blocks.
We investigate graphs that can be represented as vertex intersections of horizontal and vertical paths in a grid, the so called $B_0$-VPG graphs. Recognizing this class is an NP-complete problem. Although, there exists a polynomial time algorithm for recognizing chordal $B_0$-VPG graphs. In this paper, we present a minimal forbidden induced subgraph characterization of $B_0$-VPG graphs restricted to block graphs. As a byproduct, the proof of the main theorem provides an alternative certifying recognition and representation algorithm for $B_0$-VPG graphs in the class of block graphs.
Let C be a finite connected graph for which there is a countable universal C-free graph, and whose tree of blocks is a path. Then the blocks of C are complete. This generalizes a result of Furedi and Komjath, and fits naturally into a set of conjectures regarding the existence of countable C-free graphs, with C an arbitrary finite connected graph.
A $k$-block in a graph $G$ is a maximal set of at least $k$ vertices no two of which can be separated in $G$ by deleting fewer than $k$ vertices. The block number $β(G)$ of $G$ is the maximum integer $k$ for which $G$ contains a $k$-block. We prove a structure theorem for graphs without a $(k+1)$-block, showing that every such graph has a tree-decomposition in which every torso has at most $k$ vertices of degree $2k^2$ or greater. This yields a qualitative duality, since every graph that admits such a decomposition has block number at most $2k^2$. We also study $k$-blocks in graphs from classe…
In this note, we show that the twisted Grassmann graph constructed by van Dam and Koolen is the block graph of the design constructed by Jungnickel and Tonchev. We also show that the full automorphism group of the design is isomorphic to the full automorphism group of the twisted Grassmann graph.
Given a graph $G=(V, E)$ and and a proper labeling $f$ from $V$ to $\{1, ..., n\}$, we define $B(f)$ as the maximum absolute difference between $f(u)$ and $f(v)$ where $(u,v)\in E$. The bandwidth of $G$ is the minimum $B(f)$ for all $f$. Say $G$ is $δ$-dense if its minimum degree is $δn$. In this paper, we investigate the trade-off between the approximation ratio and the time complexity of the classical approach of Karpinski {et al}.\cite{Karpin97}, and present a faster randomized algorithm for approximating the bandwidth of $δ$-dense graphs. In particular, by removing the polylog factor of th…
A graph $G$ is \emph{nonsingular (singular)} if its adjacency matrix $A(G)$ is nonsingular (singular). In this article, we consider the nonsingularity of block graphs, i.e., graphs in which every block is a clique. Extending the problem, we characterize nonsingular vertex-weighted block graphs in terms of reduced vertex-weighted graphs resulting after successive deletion and contraction of pendant blocks. Special cases where nonsingularity of block graphs may be directly determined are discussed.
A block graph is a graph in which every block is a complete graph. Let $G$ be a block graph and let $A(G)$ be its (0,1)-adjacency matrix. Graph $G$ is called nonsingular (singular) if $A(G)$ is nonsingular (singular). An interesting open problem, proposed in 2013 by Bapat and Roy, is to characterize nonsingular block graphs. In this article, we present some classes of nonsingular and singular block graphs and related conjectures.
The infinite tumbling block graph is a bipartite graph, where each vertex in one partite set is of degree 3 and each vertex in the other partite set is of degree 6. It is a 2-dimensional array of blocks of seven vertices and nine edges, a planar graph that has 3-D looks. This paper introduces tumbling block graphs and considers various graphical parameters for different classes of infinite and finite tumbling blocks.
A radio labeling of a graph $G$ is a mapping $f$ : $V(G) \rightarrow \{0, 1, 2,...\}$ such that $|f(u)-f(v)| \geq diam(G) + 1 - d(u,v)$ holds for every pair of vertices $u$ and $v$, where $diam(G)$ is the diameter of $G$ and $d(u,v)$ is the distance between $u$ and $v$ in $G$. The radio number of $G$, denoted by $rn(G)$, is the smallest $t$ such that $G$ admits a radio labeling with $t=\max\{|f(v)-f(u)|: v, u \in V(G)\}$. A block graph is a graph such that each block (induced maximal 2-connected subgraph) is a complete graph. In this paper, a lower bound for the radio number of block graphs is…
Block graphs are graphs in which every block (biconnected component) is a clique. A graph $G=(V,E)$ is said to be an (unpartitioned) $k$-probe block graph if there exist $k$ independent sets $N_i\subseteq V$, $1\le i\le k$, such that the graph $G'$ obtained from $G$ by adding certain edges between vertices inside the sets $N_i$, $1\le i\le k$, is a block graph; if the independent sets $N_i$ are given, $G$ is called a partitioned $k$-probe block graph. In this paper we give good characterizations for $2$-probe block graphs, in both unpartitioned and partitioned cases. As an algorithmic implicat…
We compute one of the distinguished extremal Betti number of the binomial edge ideal of a block graph, and classify all block graphs admitting precisely one extremal Betti number.
We give a lower bound for the Castelnuovo-Mumford regularity of binomial edge ideals of block graphs by computing the two distinguished extremal Betti numbers of a new family of block graphs, called flower graphs. Moreover, we present a linear time algorithm to compute the Castelnuovo-Mumford regularity and Krull dimension of binomial edge ideals of block graphs.
Let $\mathcal{B}(n,q)$ be the class of block graphs on $n$ vertices having all their blocks of the same size. We prove that if $G\in \mathcal{B}(n,q)$ has at most three pairwise adjacent cut vertices then the minimum spectral radius $ρ(G)$ is attained at a unique graph. In addition, we present a lower bound for $ρ(G)$ when $G\in \mathcal{B}(n,q)$.
We find a class of block graphs whose binomial edge ideals have minimal regularity. As a consequence, we characterize the trees whose binomial edge ideals have minimal regularity. Also, we show that the binomial edge ideal of a block graph has the same depth as its initial ideal.
We consider the problem of generating uniformly random partitions of the vertex set of a graph such that every piece induces a connected subgraph. For the case where we want to have partitions with linearly many pieces of bounded size, we obtain approximate sampling algorithms based on Glauber dynamics which are fixed-parameter tractable with respect to the bandwidth of $G$, with simple-exponential dependence on the bandwidth. For example, for rectangles of constant or logarithmic width this gives polynomial-time sampling algorithms. More generally, this gives sub-exponential algorithms for bo…
