Related papers: Tiling $H$ in dense graphs
We examine the structure of higher-derivative string corrections under a cosmological reduction and make connection to generalized geometry and T-duality. We observe that, while the curvature $R^\mu{}_{\nu\rho\sigma}(\Omega_+)$ of the…
We study the uniqueness of optimal solutions to extremal graph theory problems. Lovasz conjectured that every finite feasible set of subgraph density constraints can be extended further by a finite set of density constraints so that the…
In recent years several classical results in extremal graph theory have been improved in a uniform way and their proofs have been simplified and streamlined. These results include a new Erd\H{o}s-Stone-Bollob\'as theorem, several stability…
Sidorenko's conjecture states that for every bipartite graph $H$ on $\{1,\cdots,k\}$, $\int \prod_{(i,j)\in E(H)} h(x_i, y_j) d\mu^{|V(H)|} \ge \left( \int h(x,y) \,d\mu^2 \right)^{|E(H)|}$ holds, where $\mu$ is the Lebesgue measure on…
Klimo\v{s}ov\'a, Piguet, and Rozho\v{n} conjectured that any graph with minimum degree $k/2$ and sufficiently many vertices of degree $k$ should contain all trees with $k$ edges. We prove an asymptotic version of this conjecture for dense…
A graph is near-planar if it can be obtained from a planar graph by adding an edge. We show the surprising fact that it is NP-hard to compute the crossing number of near-planar graphs. A graph is 1-planar if it has a drawing where every…
The shrinking operation converts a hypergraph into a graph by choosing, from each hyperedge, two endvertices of a corresponding graph edge. A hypertree is a hypergraph which can be shrunk to a tree on the same vertex set. Klimo\v{s}ov\'{a}…
We count tilings of a rectangle of integer sides m-1 and n-1 by a special set of tiles. The result is obtained fron the study of the kernel of the adjacency matrix of an n x n rectangular graph of Z x Z.
We show that the problem of counting perfect matchings remains #P-complete even if we restrict the input to very dense graphs, proving the conjecture in [5]. Here "dense graphs" refer to bipartite graphs of bipartite independence number…
In many proofs concerning extremal parameters of Berge hypergraphs one starts with analyzing that part of that shadow graph which is contained in many hyperedges. Capturing this phenomenon we introduce two new types of hypergraphs. A…
Recent results show that the structural similarity of graphs can be characterized by counting homomorphisms to them: the Tree Theorem states that the well-known color-refinement algorithm does not distinguish two graphs G and H if and only…
An $r$-augmented tree is a rooted tree plus $r$ edges added from each leaf to ancestors. For $d,g,r\in\mathbb{N}$, we construct a bipartite $r$-augmented complete $d$-ary tree having girth at least $g$. The height of such trees must grow…
Let $\tau(G)$ and $\tau_G(e)$ be the number of spanning trees of a connected graph $G$ and the number of spanning trees of $G$ containing edge $e$. The ratio $d_{G}(e)=\tau_{G}(e)/\tau(G)$ is called the spanning tree edge density of $e$, or…
An N -tiling of triangle ABC by triangle T is a way of writing ABC as a union of N triangles congruent to T, overlapping only at their boundaries. The triangle T is the "tile". The tile may or may not be similar to ABC . We wish to…
We construct classes of graphs that are variants of the so-called layered wheel. One of their key properties is that while the treewidth is bounded by a function of the clique number, the construction can be adjusted to make the dependance…
A variant of the Erd\H{o}s-S\'os conjecture, posed by Havet, Reed, Stein and Wood, states that every graph with minimum degree at least $\lfloor 2k/3 \rfloor$ and maximum degree at least $k$ contains a copy of every tree with $k$ edges.…
We investigate the terminal-pairibility problem in the case when the base graph is a complete bipartite graph, and the demand graph is also bipartite with the same color classes. We improve the lower bound on maximum value of $\Delta(D)$…
The family of right-angled tiling links consists of links built from regular 4-valent tilings of constant-curvature surfaces that contain one or two types of tiles. The complements of these links admit complete hyperbolic structures and…
The problem of maximising the number of cliques among $n$-vertex graphs from various graph classes has received considerable attention. We investigate this problem for the class of $1$-planar graphs where we determine precisely the maximum…
Given a graph $G$ and collection of subgraphs $T$ (called tiles), we consider covering $G$ with copies of tiles in $T$ so that each vertex $v\in G$ is covered with a predetermined multiplicity. The multinomial tiling model is a natural…