Related papers: Graphical sequences and plane trees

We first rephrase and unify known bijections between bipartite plane maps and labelled trees with the formalism of looptrees, which we argue to be both more relevant and technically simpler since the geometry of a looptree is explicitly…

We study some properties of graphs (or, rather, graph sequences) defined by demanding that the number of subgraphs of a given type, with vertices in subsets of given sizes, approximatively equals the number expected in a random graph. It…

For a labelled tree on the vertex set $[n]:=\{1,2,..., n\}$, define the direction of each edge $ij$ to be $i\to j$ if $i<j$. The indegree sequence of $T$ can be considered as a partition $\lambda \vdash n-1$. The enumeration of trees with a…

A new very simple proof of the number of labeled rooted forest-graphs with a given number of vertices is given. As a partial case of this formula we have Cayley's formula.

We study a configuration model on bipartite planar maps in which, given $n$ even integers, one samples a planar map with $n$ faces uniformly at random with these face degrees. We prove that when suitably rescaled, such maps always admit…

An order-theoretic forest is a countable partial order such that the set of elements larger than any element is linearly ordered. It is an order-theoretic tree if any two elements have an upper-bound. The order type of a branch can be any…

Let $G_1$ be a planar graph such that all cycles of length at most 4 are independent and let $G_2$ be a planar graph without 3-cycles and adjacent 4-cycles. It is proved that the set of vertices of $G_1$ and $G_2$ can be equitably…

A \emph{queue layout} of a graph consists of a \emph{linear order} of its vertices and a partition of its edges into \emph{queues}, so that no two independent edges of the same queue are nested. The \emph{queue number} of a graph is the…

This paper investigates two involutions on binary trees. One is the mirror symmetry of binary trees which combined with the classical bijection $\varphi$ between binary trees and plane trees answers an open problem posed by Bai and Chen.…

Let $F(G)$ be the number of forests of a graph $G$. Similarly let $C(G)$ be the number of connected spanning subgraphs of a connected graph $G$. We bound $F(G)$ and $C(G)$ for regular graphs and for graphs with fixed average degree. Among…

When considering the number of subtrees of trees, the extremal structures which maximize this number among binary trees and trees with a given maximum degree lead to some interesting facts that correlate to other graphical indices in…

Let $t$ be a rooted tree and $n_i(t)$ the number of nodes in $t$ having $i$ children. The degree sequence $(n_i(t),i\geq 0)$ of $t$ satisfies $\sum_{i\ge 0} n_i(t)=1+\sum_{i\ge 0} in_i(t)=|t|$, where $|t|$ denotes the number of nodes in…

We study a finite sequence of graphs, beginning with the curve graph and ending with a graph quasi-isometric to a tree. There is a Lipschitz map from one graph in the sequence to the next. This sequence was first introduced by Hamenst\"adt.…

A graph is called set-sequential if its vertices can be labeled with distinct nonzero vectors in $\mathbb{F}_2^n$ such that when each edge is labeled with the sum$\pmod{2}$ of its vertices, every nonzero vector in $\mathbb{F}_2^n$ is the…

For a given homogeneous Poisson point process in $\mathbb{R}^d$ two points are connected by an edge if their distance is bounded by a prescribed distance parameter. The behaviour of the resulting random graph, the Gilbert graph or random…

For a class $\mathcal C$ of graphs, we define $\mathcal C$-edge-brittleness of a graph $G$ as the minimum $\ell$ such that the vertex set of $G$ can be partitioned into sets inducing a subgraph in $\mathcal C$ and there are $\ell$ edges…

We show that planar graphs have bounded queue-number, thus proving a conjecture of Heath, Leighton and Rosenberg from 1992. The key to the proof is a new structural tool called layered partitions, and the result that every planar graph has…

S. B. Rao conjectured that graphic sequences are well-quasi-ordered under an inclusion based on induced subgraphs. This conjecture has now been proved by Chudnovsky and Seymour. We give an independent short proof of the labelled version of…

Degree distribution, or equivalently called degree sequence, has been commonly used to be one of most significant measures for studying a large number of complex networks with which some well-known results have been obtained. By contrast,…

We study the possible values of the matching number among all trees with a given degree sequence as well as all bipartite graphs with a given bipartite degree sequence. For tree degree sequences, we obtain closed formulas for the possible…