Related papers: 1-factorizations of pseudorandom graphs

A $1$-factor in an $n$-vertex graph $G$ is a collection of $\frac{n}{2}$ vertex-disjoint edges and a $1$-factorization of $G$ is a partition of its edges into edge-disjoint $1$-factors. Clearly, a $1$-factorization of $G$ cannot exist…

In this paper 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 $G$ on $n$…

A 1-factor of a hypergraph $G=(X,W)$ is a set of hyperedges such that every vertex of $G$ is incident to exactly one hyperedge from the set. A 1-factorization is a partition of all hyperedges of $G$ into disjoint 1-factors. The adjacency…

A 1-factorisation of a regular graph $G$ is a partition of its edge set $E(G)$ into perfect matchings of $G$. Behague asked for the minimal $r=r(d)$ such that some $1$-factorisation of the $d$-dimensional hypercube $Q_d$ has the property…

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…

The existence of $1$-factorizations of an infinite complete equipartite graph $K_m[n]$ (with $m$ parts of size $n$) admitting a vertex-regular automorphism group $G$ is known only when $n=1$ and $m$ is countable (that is, for countable…

For $d \ge 1$, $s \ge 0$ a $(d, d+s)$-{\em graph} is a graph whose degrees all lie in the interval $\{d, d+1, \ldots, d + s\}$. For $r \ge 1$, $a \ge 0$, an $(r, r+a)$-{\em factor} of a graph $G$ is a spanning $(r, r+a)$-subgraph of $G$. An…

A $1-$factorization of a complete graph on $2n$ vertices is said to be $G-$regular if it posseses an automorphism group $G$ acting sharply transitively on the vertex-set. The problem of determining which groups can realize such a situation…

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…

Given a graph $G$ and $p\in [0,1]$, the random subgraph $G_p$ is obtained by retaining each edge of $G$ independently with probability $p$. We show that for every $\epsilon>0$, there exists a constant $C>0$ such that the following holds.…

For a family $\mathcal{F}$ of graphs, a graph $G$ is called \emph{$\mathcal{F}$-universal} if $G$ contains every graph in $\mathcal{F}$ as a subgraph. Let $\mathcal{F}_n(d)$ be the family of all graphs on $n$ vertices with maximum degree at…

In 1972, Kotzig proved that for every even $n$, the complete graph $K_n$ can be decomposed into $\lceil\log_2n\rceil$ edge-disjoint regular bipartite spanning subgraphs, which is best possible. In this paper, we study regular bipartite…

A graph $G$ is $\textit{universal}$ for a (finite) family $\mathcal{H}$ of graphs if every $H \in \mathcal{H}$ is a subgraph of $G$. For a given family $\mathcal{H}$, the goal is to determine the smallest number of edges an…

Given a real $\mu\geq 1$, a graph $H$ is $\mu$-almost-regular if $\Delta(H)\leq \mu \delta(H)$. The celebrated regularization theorem of Erd\H{o}s and Simonovits states that for every real $0<\varepsilon<1$ there exists a real…

In this paper, we show that every highly edge-connected graph $G$, under a necessary and sufficient degree condition, can be edge-decomposed into $k$ factors $G_1,\ldots, G_k$ such that for each vertex $v\in V(G_i)$ with $1\le i\le k$,…

A graph $G=(V,E)$ is called $d$-rigid if, for a generic embedding of its vertices in $\mathbb{R}^d$, every edge-length preserving continuous motion of the vertices preserves the distances between all pairs of non-adjacent vertices as well.…

Kim and Vu made the following conjecture (\textit{Advances in Mathematics}, 2004): if $d\gg \log n$, then the random $d$-regular graph $\mathcal G(n,d)$ can asymptotically almost surely be "sandwiched" between $\mathcal G(n,p_1)$ and…

For unweighted graphs, finding isometric embeddings is closely related to decompositions of $G$ into Cartesian products of smaller graphs. When $G$ is isomorphic to a Cartesian graph product, we call the factors of this product a…

Let $G$ be a simple graph with maximum degree $\Delta(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>\Delta(G)\lfloor |V(H)|/2 \rfloor$. Chetwynd and Hilton in 1985 conjectured that a graph $G$ with $\Delta(G)>|V(G)|/3$ has chromatic…

We present a new approach to showing that random graphs are nearly optimal expanders. This approach is based on recent deep results in combinatorial group theory. It applies to both regular and irregular random graphs. Let G be a random…