Related papers: Sidorenko's conjecture for blow-ups

A set S of vertices in a graph G is a dominating set of G if every vertex not in S is adjacent to a vertex in S . The domination number of G, denoted by $\gamma$(G), is the minimum cardinality of a dominating set in G. In a breakthrough…

A cornerstone of extremal graph theory due to Erd\H{o}s and Stone states that the edge density which guarantees a fixed graph $F$ as subgraph also asymptotically guarantees a blow-up of $F$ as subgraph. It is natural to ask whether this…

We prove the conjecture of Seymour (1993) that for every apex-forest $H_1$ and outerplanar graph $H_2$ there is an integer $p$ such that every 2-connected graph of pathwidth at least $p$ contains $H_1$ or $H_2$ as a minor. An independent…

For every fixed graph $H$ and every fixed $0 < \alpha < 1$, we show that if a graph $G$ has the property that all subsets of size $\alpha n$ contain the ``correct'' number of copies of $H$ one would expect to find in the random graph…

Let $k,r \geq 2$ be two integers. We consider the problem of partitioning the hyperedge set of an $r$-uniform hypergraph $H$ into the minimum number $\chi_k'(H)$ of edge-disjoint subhypergraphs in which every vertex has either degree $0$ or…

Erd\H{o}s and Hajnal conjectured that for every graph $H$, there exists $c>0$ such that every $H$-free graph $G$ has a clique or a stable set of size at least $|G|^c$ (a graph is $H$-free if it has no induced subgraph isomorphic to $H$).…

A typical decomposition question asks whether the edges of some graph $G$ can be partitioned into disjoint copies of another graph $H$. One of the oldest and best known conjectures in this area, posed by Ringel in 1963, concerns the…

A graph covering projection, also referred to as a locally bijective homomorphism, is a mapping between the vertices and edges of two graphs that preserves incidences and is a local bijection. This concept originates in topological graph…

A biclique in a graph $G$ is a complete bipartite subgraph (not necessarily induced), and the least positive integer $k$ for which the vertex set of $G$ can be partitioned into at most $k$ bicliques is the biclique vertex partition number…

An \emph{$H$-packing} in a graph $G$ is a collection of pairwise vertex-disjoint copies of $H$ in $G$. We prove that for every $c > 0$ and every bipartite graph $H$, any $\lfloor cn \rfloor$-regular graph $G$ admits an $H$-packing that…

The Graph Reconstruction Conjecture famously posits that any undirected graph on at least three vertices is determined up to isomorphism by its family of (unlabeled) induced subgraphs. At present, the conjecture admits partial resolutions…

A new, constructive proof with a small explicit constant is given to the Erd\H{o}s-Pyber theorem which says that the edges of a graph on $n$ vertices can be partitioned into complete bipartite subgraphs so that every vertex is covered at…

The Kohayakawa-Nagle-R\"odl-Schacht conjecture roughly states that every sufficiently large locally $d$-dense graph $G$ on $n$ vertices must contain at least $(1-o(1))d^{|E(H)|}n^{|V(H)|}$ copies of a fixed graph $H$. Despite its important…

The 2-blowup of a graph is obtained by replacing each vertex with two non-adjacent copies; a graph is biplanar if it is the union of two planar graphs. We disprove a conjecture of Gethner that 2-blowups of planar graphs are biplanar:…

Let $G = (V, E)$ be a graph and $\sigma(G)$ the number of independent (vertex) sets in $G$. Then the Merrifield-Simmons conjecture states that the sign of the term $\sigma(G_{-u}) \cdot \sigma(G_{-v}) - \sigma(G) \cdot \sigma(G_{-u-v})$…

The bipartite-hole-number of a graph $G$, denoted by $\widetilde{\alpha}(G)$, is the minimum number $k$ such that there exist positive integers $s$ and $t$ with $s+t=k+1$ with the property that for any two disjoint sets $A,B\subseteq V(G)$…

We prove that for every graph $H$, there exists $\varepsilon>0$ such that every $n$-vertex graph with no vertex-minors isomorphic to $H$ has a pair of disjoint sets $A$, $B$ of vertices such that $|A|, |B|\ge \varepsilon n$ and $A$ is…

The Erd\H{o}s-Hajnal Conjecture states that for every $H$ there exists a constant $\epsilon(H)>0$ such that every graph $G$ that does not contain $H$ as an induced subgraph contains a clique or a stable set of size at least…

Ryser's conjecture says that for every $r$-partite hypergraph $H$ with matching number $\nu(H)$, the vertex cover number is at most $(r-1)\nu(H)$. This far reaching generalization of K\"onig's theorem is only known to be true for $r\leq 3$,…

A famous conjecture by Itai and Zehavi states that, for every $d$-vertex-connected graph $G$ and every vertex $r$ in $G$, there are $d$ spanning trees of $G$ such that, for every vertex $v$ in $G\setminus \{r\}$, the paths between $r$ and…