Related papers: The neighborhood complex of a random graph

Each graphon $W:\Omega^2\rightarrow[0,1]$ yields an inhomogeneous random graph model $G(n,W)$. We show that $G(n,W)$ is asymptotically almost surely connected if and only if (i) $W$ is a connected graphon and (ii) the measure of elements of…

A simplicial complex is $r$-conic if every subcomplex of at most $r$ vertices is contained in the star of a vertex. A $4$-conic complex is simply connected. We prove that an $8$-conic complex is $2$-connected. In general a $(2n+1)$-conic…

Complexes of (not) connected graphs, hypergraphs and their homology appear in the construction of knot invariants given by V. Vassiliev. In this paper we study the complexes of not $i$-connected $k$-hypergraphs on $n$ vertices. We show that…

Let $H$ and $G$ be graphs such that $H$ has at least 3 vertices and is connected. The $H$-line graph of $G$, denoted by $HL(G)$, is that graph whose vertices are the edges of $G$ and where two vertices of $HL(G)$ are adjacent if they are…

To what extent is it possible to visualize high-dimensional data in two- or three-dimensional plots? We reframe this question in terms of embedding $n$-vertex graphs (representing the neighborhood structure of the input points) into metric…

Let $k\ge 2$ be fixed integer, $0<c<1$ a constant. Consider a graph $G$ with $n$ vertices and average degree $cn$. We answer a question of Simon Griffiths by showing that $G$ has $k$ vertices such that their neighborhoods together cover at…

Let $d,n\in \mathbb{N}$ be such that $d=\omega(1)$, and $d\le n^{1-a}$ for some constant $a>0$. Consider a $d$-regular graph $G=(V, E)$ and the random graph process that starts with the empty graph $G(0)$ and at each step $G(i)$ is obtained…

An (m,n)-colored mixed graph, or simply, an (m,n)-graph is a graph having m different types of arcs and n different types of edges. A homomorphism of an (m,n)-graph G to another (m,n)-graph H is a vertex mapping that preserves adjacency,…

A path in an edge-colored graph is called a \emph{rainbow path} if all edges on it have pairwise distinct colors. For $k\geq 1$, the \emph{rainbow-$k$-connectivity} of a graph $G$, denoted $rc_k(G)$, is the minimum number of colors required…

A random algebraic graph is defined by a group $G$ with a uniform distribution over it and a connection $\sigma:G\longrightarrow[0,1]$ with expectation $p,$ satisfying $\sigma(g)=\sigma(g^{-1}).$ The random graph…

We show that the ratio of the number of near perfect matchings to the number of perfect matchings in $d$-regular strong expander (non-bipartite) graphs, with $2n$ vertices, is a polynomial in $n$, thus the Jerrum and Sinclair Markov chain…

In this paper, we study the homology of the coloring complex and the cyclic coloring complex of a complete $k$-uniform hypergraph. We show that the coloring complex of a complete $k$-uniform hypergraph is shellable, and we determine the…

Kierstead, Szemer\'edi, and Trotter showed that a graph with at most $\lfloor r/(2n)\rfloor^n$ vertices such that each ball of radius $r$ in it is $c$-colorable should have chromatic number at most $n(c-1)+1$. We show that this estimate is…

Mossel and Ross (2019) introduce the shotgun assembly problem for random graphs: what radius $R$ ensures that the random graph $G$ can be uniquely recovered from its list of rooted $R$-neighborhoods, with high probability? Here we consider…

The "clustered chromatic number" of a class of graphs is the minimum integer $k$ such that for some integer $c$ every graph in the class is $k$-colourable with monochromatic components of size at most $c$. We prove that for every graph $H$,…

We study random graphs with possibly different edge probabilities in the challenging sparse regime of bounded expected degrees. Unlike in the dense case, neither the graph adjacency matrix nor its Laplacian concentrate around their…

Consider a graph $G$ with chromatic number $k$ and a collection of complete bipartite graphs, or bicliques, that cover the edges of $G$. We prove the following two results: \medskip \noindent $\bullet$ If the bicliques partition the edges…

Finding the largest clique is a notoriously hard problem, even on random graphs. It is known that the clique number of a random graph G(n,1/2) is almost surely either k or k+1, where k = 2log n - 2log(log n) - 1. However, a simple greedy…

A graph H is k-common if the number of monochromatic copies of H in a k-edge-coloring of K_n is asymptotically minimized by a random coloring. For every k, we construct a connected non-bipartite k-common graph. This resolves a problem…

Let $G$ be a $k$ - connected ($k \geq 2$) graph of order $n$. If $\chi(G) \geq n - k$, then $G$ is Hamiltonian or $K_k \vee (K_k^c \cup K_{n - 2k})$ with $n \geq 2 k + 1$, where $\chi(G)$ is the chromatic number of the graph $G$.