Related papers: 1-factorizations of pseudorandom graphs

We provide sufficient conditions for a regular graph $G$ of growing degree $d$, guaranteeing a phase transition in its random subgraph $G_p$ similar to that of $G(n,p)$ when $p\cdot d\approx 1$. These conditions capture several well-studied…

We construct a fixed parameter algorithm parameterized by d and k that takes as an input a graph G' obtained from a d-degenerate graph G by complementing on at most k arbitrary subsets of the vertex set of G and outputs a graph H such that…

Let $G=(V,E)$ be a simple connected graph. A matching of $G$ is a set of disjoint edges of $G$. For every $n, m\in\mathbb{N}$, the $n$-subdivision of $G$ is a simple graph $G^{\frac{1}{n}}$ which is constructed by replacing each edge of $G$…

We say that a $d$-regular graph is a $\gamma$-expander if for every not too large set of vertices $S$, there are at least $\gamma d |S|$ edges leaving $S$, and we say that a graph $G$ is $\gamma$-far from bipartite if at least $\gamma e(G)$…

A finite non-increasing sequence of positive integers $d = (d_1\geq \cdots\geq d_n)$ is called a degree sequence if there is a graph $G = (V,E)$ with $V = \{v_1,\ldots,v_n\}$ and $deg(v_i)=d_i$ for $i=1,\ldots,n$. In that case we say that…

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…

We describe a new random greedy algorithm for generating regular graphs of high girth: Let $k\geq 3$ and $c \in (0,1)$ be fixed. Let $n \in \mathbb{N}$ be even and set $g = c \log_{k-1} (n)$. Begin with a Hamilton cycle $G$ on $n$ vertices.…

In data transmission networks, the availability of data transmission is equivalent to the existence of the fractional factor of the corresponding graph which is generated by the network. Research on the existence of fractional factors under…

Extremal problems involving the enumeration of graph substructures have a long history in graph theory. For example, the number of independent sets in a $d$-regular graph on $n$ vertices is at most $(2^{d+1}-1)^{n/2d}$ by the Kahn-Zhao…

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by no more than one other edge (and any pair of crossing edges cross only once). A non-1-planar graph $G$ is minimal if the graph $G-e$ is 1-planar for every…

An one-factorization $\mathcal{F}$ of the complete graph $K_n$ is ($l,C_k$), where $l\geq0$ and $k\geq4$ are integers, if the union $F\cup G$, for any $F,G\in\mathcal{F}$, includes exactly $l$ (edge-disjoint) cycles of length $k$ ($lk\leq…

For a graph $G=(V,E)$, let $\tau(G)$ denote the minimum number of pairwise edge disjoint complete bipartite subgraphs of $G$ so that each edge of $G$ belongs to exactly one of them. It is easy to see that for every graph $G$, $\tau(G) \leq…

We consider the problem of decomposing the edges of a directed graph into as few paths as possible. There is a natural lower bound for the number of paths needed in an edge decomposition of a directed graph $D$ in terms of its degree…

A {\it fractional matching} of a graph $G$ is a function $f$ giving each edge a number in $[0,1]$ so that $\sum_{e \in \Gamma(v)} f(e) \le 1$ for each $v\in V(G)$, where $\Gamma(v)$ is the set of edges incident to $v$. The {\it fractional…

Let $d\geq 3$ be fixed and $G$ be a large random $d$-regular graph on $n$ vertices. We show that if $n$ is large enough then the entry distribution of every almost eigenvector $v$ of $G$ (with entry sum 0 and normalized to have length…

Let $K^r_n$ be the complete $r$-uniform hypergraph on $n$ vertices, that is, the hypergraph whose vertex set is $[n]:=\{1,2,...,n\}$ and whose edge set is $\binom{[n]}{r}$. We form $G^r(n,p)$ by retaining each edge of $K^r_n$ independently…

For a graph $G=(V,E)$ with $v(G)$ vertices the partition function of the random cluster model is defined by $$Z_G(q,w)=\sum_{A\subseteq E(G)}q^{k(A)}w^{|A|},$$ where $k(A)$ denotes the number of connected components of the graph $(V,A)$.…

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…

For a real c \geq 1 and an integer n, let f(n,c) denote the maximum integer f so that every graph on n vertices contains an induced subgraph on at least f vertices in which the maximum degree is at most c times the minimum degree. Thus, in…

We prove a `resilience' version of Dirac's theorem in the setting of random regular graphs. More precisely, we show that, whenever $d$ is sufficiently large compared to $\varepsilon>0$, a.a.s. the following holds: let $G'$ be any subgraph…