Related papers: Graph norms and Sidorenko's conjecture

The symmetric difference of two graphs $G_1,G_2$ on the same set of vertices $V$ is the graph on $V$ whose set of edges are all edges that belong to exactly one of the two graphs $G_1,G_2$. For a fixed graph $H$ call a collection ${\cal G}$…

Graph symmetries intervene in diverse applications, from enumeration, to graph structure compression, to the discovery of graph dynamics (e.g., node arrival order inference). Whereas Erd\H{o}s-R\'enyi graphs are typically asymmetric, real…

In this article we study graphs with ordering of vertices, we define a generalization called a pseudoordering, and for a graph $H$ we define the $H$-Hamiltonian number of a graph $G$. We will show that this concept is a generalization of…

We investigate the parameterized complexity of the recognition problem for the proper $H$-graphs. The $H$-graphs are the intersection graphs of connected subgraphs of a subdivision of a multigraph $H$, and the properness means that the…

Given a fixed graph H, we say that a graph G is H-free if G does not contain H as a subgraph. The Tur\'an number ex(n, H) of H is the maximum number of edges in an n-vertex H-free graph. The study of Tur\'an number of graphs is a central…

Hamiltonian cycles in graphs were first studied in the 1850s. Since then, an impressive amount of research has been dedicated to identifying classes of graphs that allow Hamiltonian cycles, and to related questions. The corresponding…

Hadwiger's Conjecture states that every graph with chromatic number $k$ contains a complete graph on $k$ vertices as a minor. This conjecture is a tremendous strengthening of the Four-Colour Theorem and is regarded as one of the most…

Given a fixed graph $H$, a real number $p\in(0,1)$, and an infinite Erd\H{o}s-R\'enyi graph $G\sim G(\infty,p)$, how many adjacency queries do we have to make to find a copy of $H$ inside $G$ with probability $1/2$? Determining this number…

In this paper, we derive nearly tight probabilistic norm bounds for a class of random matrices we call graph matrices. While the classical case of symmetric matrices with independent random entries (Wigner's matrices) is a special case, in…

Ramsey's Theorem guarantees for every graph H that any 2-edge-coloring of a sufficiently large complete graph contains a monochromatic copy of H. In 1962, Erdos conjectured that the random 2-edge-coloring minimizes the number of…

A homomorphism from a graph $G$ to a graph $H$ is an edge-preserving mapping from $V(G)$ to $V(H)$. For a fixed graph $H$, in the list homomorphism problem, denoted by LHom($H$), we are given a graph $G$, whose every vertex $v$ is equipped…

Counting small patterns in a large dataset is a fundamental algorithmic task. The most common version of this task is subgraph/homomorphism counting, wherein we count the number of occurrences of a small pattern graph $H$ in an input graph…

Hoffman's bound is a well-known spectral bound on the chromatic number of a graph, known to be tight for instance for bipartite graphs. While Hoffman colorings (colorings attaining the bound) were studied before for regular graphs, for…

Homomorphism is a key mapping technique between graphs that preserves their structure. Given a graph and a pattern, the subgraph homomorphism problem involves finding a mapping from the pattern to the graph, ensuring that adjacent vertices…

Let $G= (V,E)$ be a finite graph. For $d_0>0$ we say that $G$ is $d_0$-regular, if every $v\in V$ has degree $d_0$. We say that $G$ is $(d_0, d_1)$-regular, for $0<d_1<d_0$, if $G$ is $d_0$ regular and for every $v\in V$, the subgraph…

A graph $G$ admits an $H$-tiling if it contains a collection of vertex-disjoint copies of $H$. In this paper, we confirm a conjecture proposed by K\"{u}hn, Osthus, and Treglown by showing that for any given graph $H$, there exists a…

Let $n \equiv 0\, (\, \text{mod } 3\,)$ and $H_{n, n/3}^2$ be the 3-graph of order $n$, whose vertex set is partitioned into two sets $S$ and $T$ of size $\frac{1}{3}n+1$ and $\frac{2}{3}n -1$, respectively, and whose edge set consists of…

In this paper we study the following problem. Let $A$ be a fixed graph, and let $\hom(G,A)$ denote the number of homomorphisms from a graph $G$ to $A$. Furthermore, let $v(G)$ denote the number of vertices of $G$, and let $\mathcal{G}_d$…

We introduce graph motif parameters, a class of graph parameters that depend only on the frequencies of constant-size induced subgraphs. Classical works by Lov\'asz show that many interesting quantities have this form, including, for fixed…

In this paper, we extend two classical results about the density of subgraphs of hypercubes to subgraphs $G$ of Cartesian products $G_1\times\cdots\times G_m$ of arbitrary connected graphs. Namely, we show that $\frac{|E(G)|}{|V(G)|}\le…