Related papers: A simple and sharper proof of the hypergraph Moore…
Sidorenko's conjecture states that, for all bipartite graphs $H$, quasirandom graphs contain asymptotically the minimum number of copies of $H$ taken over all graphs with the same order and edge density. While still open for graphs, the…
A central problem in extremal graph theory is to estimate, for a given graph $H$, the number of $H$-free graphs on a given set of $n$ vertices. In the case when $H$ is not bipartite, fairly precise estimates on this number are known. In…
In a recent work [2] with Datta, we introduced the mu vector (with respect to a given field) of simplicial complexes and used it to study tightness and lower bounds. In this paper, we modify the definition of mu vectors. With the new…
In this paper, we study the maximum order $v(k,\theta)$ of a connected $k$-regular graph whose second largest eigenvalue is at most $\theta$. From Alon-Boppana and Serre, we know that $v(k,\theta)$ is finite when $\theta < 2\sqrt{k-1}$…
Let $G$ be a finite connected graph, and let $T$ be a spanning tree of $G$ chosen uniformly at random. The work of Kirchhoff on electrical networks can be used to show that the events $e_1 \in T$ and $e_2 \in T$ are negatively correlated…
A classical result of Koml\'os, S\'ark\"ozy and Szemer\'edi shows that every large $n$-vertex graph with minimum degree at least $(1/2+\gamma)n$ contains all spanning trees of bounded degree. We generalised this result to loose spanning…
We study the question of the least number of random edges that need to be added to a P\'osa-Seymour graph, that is, a graph with minimum degree exceeding $\frac k{k+1}n$, to secure the existence of the $m$-th power of a Hamiltonian cycle,…
The edge domination number $\gamma_e(G)$ of a graph $G$ is the minimum size of a maximal matching in $G$. It is well known that this parameter is computationally very hard, and several approximation algorithms and heuristics have been…
For integers $k$ and $\ell$, let $\operatorname{ind}(k, \ell)$ be the maximum proportion of $k$-vertex subsets of a large graph that induce exactly $\ell$ edges. The edge-statistics theorem (conjectured by Alon-Hefetz-Krivelevich-Tyomkyn,…
Let $T$ be a tree on $t$ vertices. We prove that for every positive integer $k$ and every graph $G$, either $G$ contains $k$ pairwise vertex-disjoint subgraphs each having a $T$ minor, or there exists a set $X$ of at most $t(k-1)$ vertices…
Approximating the graph diameter is a basic task of both theoretical and practical interest. A simple folklore algorithm can output a 2-approximation to the diameter in linear time by running BFS from an arbitrary vertex. It has been open…
A family of graphs $\mathcal{F}$ is $H$-intersecting if the edge intersection of any two graphs in $\mathcal{F}$ contains a copy of a fixed graph $H$. A fundamental problem is to determine the maximum size of such a family. The trivial…
Consider a graph with n nodes and m edges, independent edge weights and lengths, and arbitrary distance demands for node pairs. The spanner problem asks for a minimum-weight subgraph that satisfies these demands via sufficiently short paths…
Random K-out graphs are used in several applications including modeling by sensor networks secured by the random pairwise key predistribution scheme, and payment channel networks. The random K-out graph with $n$ nodes is constructed as…
The Linear Arboricity Conjecture asserts that the linear arboricity of a graph with maximum degree $\Delta$ is $\lceil (\Delta+1)/2 \rceil$. For a $2k$-regular graph $G$, this implies $la(G) = k+1$. In this note, we utilize a network flow…
A well-known conjecture of F\"uredi, Kahn, and Seymour (1993) on non-uniform hypergraph matching states that for any hypergraph with edge weights $w$, there exists a matching $M$ such that the inequality $\sum_{e\in M} g(e) w(e) \geq…
A well-known conjecture of Erd\H{o}s and S\'os states that every graph with average degree exceeding $m-1$ contains every tree with $m$ edges as a subgraph. We propose a variant of this conjecture, which states that every graph of maximum…
In this paper, we prove the conjectures of Gharakhloo and Welker (2023) that the positive matching decomposition number (pmd) of a $3$-uniform hypergraph is bounded from above by a polynomial of degree $2$ in terms of the number of…
A topological graph is \emph{$k$-quasi-planar} if it does not contain $k$ pairwise crossing edges. A topological graph is \emph{simple} if every pair of its edges intersect at most once (either at a vertex or at their intersection). In…
Let $f_r(n,v,e)$ denote the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices, in which the union of any $e$ distinct edges contains at least $v+1$ vertices. The study of $f_r(n,v,e)$ was initiated by Brown, Erd{\H{o}}s…