Related papers: A proof of the Upper Matching Conjecture for large…

Motivated by the scaling limits of the connected components of the configuration model, we study uniform connected multigraphs with fixed degree sequence $\mathcal{D}$ and with surplus $k$. We call those random graphs…

Answering questions of Y. Rabinovich, we prove "stability" versions of upper bounds on maximal independent set counts in graphs under various restrictions. Roughly these say that being close to the maximum implies existence of a large…

In this paper, we make progress on a question related to one of Galvin that has attracted substantial attention recently. The question is that of determining among all graphs $G$ with $n$ vertices and $\Delta(G)\leq r$, which has the most…

There has been interest recently in maximizing the number of independent sets in graphs. For example, the Kahn-Zhao theorem gives an upper bound on the number of independent sets in a $d$-regular graph. Similarly, it is a corollary of the…

For a fixed graph G, a maximal independent set is an independent set that is not a proper subset of any other independent set. P. Erd\"os, and independently, J. W. Moon and L. Moser, and R. E. Miller and D. E. Muller, determined the maximum…

Consider the random process in which the edges of a graph $G$ are added one by one in a random order. A classical result states that if $G$ is the complete graph $K_{2n}$ or the complete bipartite graph $K_{n,n}$, then typically a perfect…

Given an $n\times n$ symmetric matrix $W\in [0,1]^{[n]\times [n]}$, let $\mathcal{G}(n,W)$ be the random graph obtained by independently including each edge $jk$ with probability $W_{jk}$. Given a degree sequence ${\bf d}=(d_1,\ldots,…

We prove that, for every integer $d$ with $d\geq 3$, there is an approximation algorithm for the maximum induced matching problem restricted to $\{ C_3,C_5\}$-free $d$-regular graphs with performance ratio $0.708\bar{3}d+0.425$, which…

In 1979 Frankl conjectured that in a finite non-trivial union-closed collection of sets there has to be an element that belongs to at least half the sets. We show that this is equivalent to the conjecture that in a finite non-trivial graph…

We study the connection between the degree sequence of a $k$-uniform hypergraph and the size of its largest matching. Let $\mathcal{F}$ be a $k$-uniform hypergraph on $n$ vertices and let $d_1 \ge d_2 \ge \dots \ge d_n$ be the vertex…

In a sequence of four papers, we prove the following results (via a unified approach) for all sufficiently large $n$: (i) [1-factorization conjecture] Suppose that $n$ is even and $D\geq 2\lceil n/4\rceil -1$. Then every $D$-regular graph…

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…

In 1965 Erd\H{o}s conjectured that the number of edges in k-uniform hypergraphs on n vertices in which the largest matching has s edges is maximized for hypergraphs of one of two special types. We settled this conjecture in the affirmative…

We prove almost sure convergence of the maximum degree in an evolving graph model combining a growing number of local choices with sublinear preferential attachment. At each step in the growth of the graph, a new vertex is introduced. Then…

We determine the maximum number of maximal independent sets of arbitrary graphs in terms of their covering numbers and we completely characterize the extremal graphs. As an application, we give a similar result for K\"onig-Egerv\'ary graphs…

We prove essentially sharp bounds for Ramsey numbers of ordered hypergraph matchings, inroduced recently by Dudek, Grytczuk, and Ruci\'{n}ski. Namely, for any $r \ge 2$ and $n \ge 2$, we show that any collection $\mathcal H$ of $n$ pairwise…

Let $n, d$ be integers with $1 \leq d \leq \left \lfloor \frac{n-1}{2} \right \rfloor$, and set $h(n,d):={n-d \choose 2} + d^2$. Erd\H{o}s proved that when $n \geq 6d$, each nonhamiltonian graph $G$ on $n$ vertices with minimum degree…

The famous List Colouring Conjecture from the 1970s states that for every graph $G$ the chromatic index of $G$ is equal to its list chromatic index. In 1996 in a seminal paper, Kahn proved that the List Colouring Conjecture holds…

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.…

The Kruskal--Katona theorem determines the maximum number of $d$-cliques in an $n$-edge $(d-1)$-uniform hypergraph. A generalization of the theorem was proposed by Bollob\'as and Eccles, called the partial shadow problem. The problem asks…