Related papers: Increasing Trees and the Degree-Chromatic Polynomi…
We consider a generalisation of the classical Ramsey theory setting to a setting where each of the edges of the underlying host graph is coloured with a {\em set} of colours (instead of just one colour). We give bounds for monochromatic…
Trees or rooted trees have been generously studied in the literature. A forest is a set of trees or rooted trees. Here we give recurrence relations between the number of some kind of rooted forest with $k$ roots and that with $k+1$ roots on…
Mayer's theory of cluster integrals allows one to write the partition function of a gas model as a generating function of weighted graphs. Recently, Labelle, Leroux and Ducharme have studied the graph weights arising from the…
Floor diagrams are a class of weighted oriented graphs introduced by E. Brugalle and the second author. Tropical geometry arguments lead to combinatorial descriptions of (ordinary and relative) Gromov-Witten invariants of projective spaces…
We define some new sequences of recursively constructed random combinatorial trees, and show that, after properly rescaling graph distance and equipping the trees with the uniform measure on vertices, each sequence converges almost surely…
We discuss a recursive formula for number of spanning trees in a graph. The paper is written primary for school students.
Let G be a graph with vertex set {1,...,n}. A spanning forest F of G is increasing if the sequence of labels on any path starting at the minimum vertex of a tree of F form an increasing sequence. Hallam and Sagan showed that the generating…
We introduce a new notation for representing labeled regular bipartite graphs of arbitrary degree. Several enumeration problems for labeled and unlabeled regular bipartite graphs have been introduced. A general algorithm for enumerating all…
In biology, a phylogenetic tree is a tool to represent the evolutionary relationship between species. Unfortunately, the classical Schr\"oder tree model is not adapted to take into account the chronology between the branching nodes. In…
We obtain new results for the probabilistic model introduced in Menshikov et al (2007) and Volkov (2006) which involves a $d$-ary regular tree. All vertices are coloured in one of $d$ distinct colours so that $d$ children of each vertex all…
Suppose $G$ is a tree. Graham's "Tree Reconstruction Conjecture" states that $G$ is uniquely determined by the integer sequence $|G|$, $|L(G)|$, $|L(L(G))|$, $|L(L(L(G)))|$, $\ldots$, where $L(H)$ denotes the line graph of the graph $H$.…
We prove several bounds on the number of incidences between two sets of multivariate polynomials of bounded degree over finite fields. From these results, we deduce bounds on incidences between points and multivariate polynomials, extending…
We study a family of tree-type diagrams that arise in studies of the cumulant expansion in discrete Erd\H os-R\'enyi random matrix models. Using a version of the Pr\" ufer code, we obtain an explicit expression for the number of tree-type…
The Brownian continuum tree was extensively studied in the 90s as a universal random metric space. One construction obtains the continuum tree by a change of metric from an excursion function (or continuous circle mapping) on $[0,1]$. This…
Given a finite planar graph, a grove is a spanning forest in which every component tree contains one or more of a specified set of vertices (called nodes) on the outer face. For the uniform measure on groves, we compute the probabilities of…
In this paper we continue the study of permutations avoiding the vincular pattern $1-32-4$ by constructing a generating tree with a single label for these permutations. This construction finally provides a clearer explanation of why a…
In the Properly Colored Spanning Tree problem, we are given an edge-colored undirected graph and the goal is to find a properly colored spanning tree, i.e., a spanning tree in which any two adjacent edges have distinct colors. The problem…
We investigate adjacency labeling schemes for graphs of bounded degree $\Delta = O(1)$. In particular, we present an optimal (up to an additive constant) $\log n + O(1)$ adjacency labeling scheme for bounded degree trees. The latter scheme…
We study a model of random binary trees grown "by the leaves" in the style of Luczak and Winkler. If $\tau_n$ is a uniform plane binary tree of size $n$, Luczak and Winkler, and later explicitly Caraceni and Stauffer, constructed a measure…
Generalizing the classical matrix-tree theorem we provide a formula counting subgraphs of a given graph with a fixed 2-core. We use this generalization to obtain an analog of the matrix-tree theorem for the root system $D_n$ (the classical…