Related papers: Enumeration of labelled 4-regular planar graphs II…

A sharp asymptotic formula for the number of strongly connected digraphs on $n$ labelled vertices with $m$ arcs, under a condition $m-n\to\infty$, $m=O(n)$, is obtained; this solves a problem posed by Wright back in $1977$. Our formula is a…

We consider the set of all graphs on n labeled vertices with prescribed degrees D=(d_1, ..., d_n). For a wide class of tame degree sequences D we prove a computationally efficient asymptotic formula approximating the number of graphs within…

A 4-regular planar graph $G$ is said to be circle representable if there exists a collection of circles drawn on the plane such that the touching and crossing points correspond to the vertices of $G$, and the circular arcs between those…

In this paper we relate a fundamental parameter of a random graph, its degree sequence, to a simple model of nearly independent binomial random variables. This confirms a conjecture made in 1997. As a result, many interesting functions of…

A matchstick graph is a planar unit-distance graph. We call it \emph{4-regular} if every vertex has degree 4. While examples of 4-regular matchstick graphs with fewer than 63 vertices are known only for $n \in \{52, 54, 57, 60\}$, we prove…

We derive an asymptotic formula for the number of connected 3-uniform hypergraphs with vertex set $[N]$ and $M$ edges for $M=N/2+R$ as long as $R$ satisfies $R = o(N)$ and $R=\omega(N^{1/3}\ln^{2} N)$. This almost completely fills the gap…

We prove a formula for the asymptotic number of edge-colored regular graphs with a prescribed set of allowed vertex-incidence structures. The formula depends on specific critical points of a polynomial encoding the vertex-incidences. As an…

Computing the embedding distribution of a given graph is a fundamental question in topological graph theory. In this article, we extend our viewpoint to a sequence of graphs and consider their asymptotic embedding distributions, which are…

We present the first parallel fixed-parameter algorithm for subgraph isomorphism in planar graphs, bounded-genus graphs, and, more generally, all minor-closed graphs of locally bounded treewidth. Our randomized low depth algorithm has a…

Let P_{n,m} denote the graph taken uniformly at random from the set of all planar graphs on {1,2,..., n} with exactly m(n) edges. We use counting arguments to investigate the probability that P_{n,m} will contain given components and…

Using three supercomputers, we broke a record set in 2011, in the enumeration of non-isomorphic regular graphs by expanding the sequence of A006820 in Online Encyclopedia of Integer Sequences (OEIS), to achieve the number for 4-regular…

An \textit{$(n,m)$-graph} $G$ is a graph having both arcs and edges, and its arcs (resp., edges) are labeled using one of the $n$ (resp., $m$) different symbols. An \textit{$(n,m)$-complete graph} $G$ is an $(n,m)$-graph without loops or…

In this paper we enumerate $k$-noncrossing tangled-diagrams. A tangled-diagram is a labeled graph whose vertices are $1,...,n$ have degree $\le 2$, and are arranged in increasing order in a horizontal line. Its arcs are drawn in the upper…

A hypergraph is simple if it has no loops and no repeated edges, and a hypergraph is linear if it is simple and each pair of edges intersects in at most one vertex. For $n\geq 3$, let $r= r(n)\geq 3$ be an integer and let $\boldsymbol{k} =…

For the set of graphs with a given degree sequence, consisting of any number of $2's$ and $1's$, and its subset of bipartite graphs, we characterize the optimal graphs who maximize and minimize the number of $m$-matchings. We find the…

We enumerate the connected graphs that contain a linear number of edges with respect to the number of vertices. So far, only the first term of the asymptotics was known. Using analytic combinatorics, i.e. generating function manipulations,…

Let $d \geq 3$ be a fixed integer. We give an asympotic formula for the expected number of spanning trees in a uniformly random $d$-regular graph with $n$ vertices. (The asymptotics are as $n\to\infty$, restricted to even $n$ if $d$ is…

For a simple finite graph G denote by {G \brace k} the number of ways of partitioning the vertex set of G into k non-empty independent sets (that is, into classes that span no edges of G). If E_n is the graph on n vertices with no edges…

This paper establishes an upper bound on the number of generalized cospectral mates of simple graphs, where the generalized spectrum consists of the spectrum of a graph and its complement. Moving beyond the classical problem of identifying…

An octilinear drawing of a planar graph is one in which each edge is drawn as a sequence of horizontal, vertical and diagonal at 45 degrees line-segments. For such drawings to be readable, special care is needed in order to keep the number…