Let $F=\{H_1,...,H_k\}$ be a family of graphs. A graph $G$ with $m$ edges is called {\em totally $F$-decomposable} if for {\em every} linear combination of the form $α_1 e(H_1) + ... + α_k e(H_k) = m$ where each $α_i$ is a nonnegative integer, there is a coloring of the edges of $G$ with $α_1+...+α_k$ colors such that exactly $α_i$ color classes induce each a copy of $H_i$, for $i=1,...,k$. We prove that if $F$ is any fixed family of trees then $\log n/n$ is a sharp threshold function for the property that the random graph $G(n,p)$ is totally $F$-decomposable. In particular, if $H$ is a tree, then $\log n/n$ is a sharp threshold function for the property that $G(n,p)$ contains $\lfloor e(G)/e(H) \rfloor$ edge-disjoint copies of $H$.
In this work we introduce and study various generalizations of the notion of increasingly labelled trees, where the label of a child node is always larger than the label of its parent node, to multilabelled tree families, where the nodes in the tree can get multiple labels. For all tree classes we show characterizations of suitable generating functions for the tree enumeration sequence via differential equations. Furthermore, for several combinatorial classes of multilabelled increasing tree families we present explicit enumeration results. We also present multilabelled increasing tree familie…
We consider so-called simple families of labelled trees, which contain, e.g., ordered, unordered, binary and cyclic labelled trees as special instances, and study the global and local behaviour of the number of inversions. In particular we obtain limiting distribution results for the total number of inversions as well as the number of inversions induced by the node labelled j in a random tree of size n.
An old conjecture of Ringel states that every tree with $m$ edges decomposes the complete graph $K_{2m+1}$. The best known lower bound for the order of a complete graph which admits a decomposition by every given tree with $m$ edges is $O(m^3)$. We show that asymptotically almost surely a random tree with $m$ edges and $p=2m+1$ a prime decomposes $K_{2m+1}(r)$ for every $r\ge 2$, the graph obtained from the complete graph $K_{2m+1}$ by replacing each vertex by a coclique of order $r$. Based on this result we show, among other results, that a random tree with $m+1$ edges a.a.s. decomposes the c…
It is known that any planar graph with diameter D has treewidth O(D), and this fact has been used as the basis for several planar graph algorithms. We investigate the extent to which similar relations hold in other graph families. We show that treewidth is bounded by a function of the diameter in a minor-closed family, if and only if some apex graph does not belong to the family. In particular, the O(D) bound above can be extended to bounded-genus graphs. As a consequence, we extend several approximation algorithms and exact subgraph isomorphism algorithms from planar graphs to other graph fam…
Let ${\cal T}=(T,w)$ be a weighted finite tree with leaves $1,..., n$. For any $I :=\{i_1,..., i_k \} \subset \{1,...,n\}$, let $D_I ({\cal T})$ be the weight of the minimal subtree of $T$ connecting $i_1,..., i_k$; the $D_{I} ({\cal T})$ are called $k$-weights of ${\cal T}$. Given a family of real numbers parametrized by the $k$-subsets of $\{1,..., n\}$, $\{D_I\}_{I \in {\{1,...,n\} \choose k}}$, we say that a weighted tree ${\cal T}=(T,w)$ with leaves $1,..., n$ realizes the family if $D_I({\cal T})=D_I$ for any $ I $. We give a characterization of the families of real numbers that are real…
Bouttier, Di Francesco and Guitter introduced a method for solving certain classes of algebraic recurrence relations arising the context of embedded trees and map enumeration. The aim of this note is to apply this method to three problems. First, we discuss a general family of embedded binary trees, trying to unify and summarize several enumeration results for binary tree families, and also to add new results. Second, we discuss the family of embedded $d$-ary trees, embedded in the plane in a natural way. Third, we show that several enumeration problems concerning simple families of lattice pa…
We extend classical results on simple varieties of trees (asymptotic enumeration, average behavior of tree parameters) to trees counted by their number of leaves. Motivated by genome comparison of related species, we then apply these results to strong interval trees with a restriction on the arity of prime nodes. Doing so, we describe a filtration of the set of permutations based on their strong interval trees. This filtration is also studied from a purely analytical point of view, thus illustrating the convergence of analytic series towards a non-analytic limit at the level of the asymptotic …
Given a tree $T$, let $q(T)$ be the minimum number of distinct eigenvalues in a symmetric matrix whose underlying graph is $T$. It is well known that $q(T)\geq d(T)+1$, where $d(T)$ is the diameter of $T$, and a tree $T$ is said to be diminimal if $q(T)=d(T)+1$. In this paper, we present families of diminimal trees of any fixed diameter. Our proof is constructive, allowing us to compute, for any diminimal tree $T$ of diameter $d$ in these families, a symmetric matrix $M$ with underlying graph $T$ whose spectrum has exactly $d+1$ distinct eigenvalues.
A symbolic-computational algorithm, fully implemented in Maple, is described, that computes explicit expressions for generating functions that enable the efficient computations of the expectation, variance, and higher moments, of the random variable `sum of distances to the root', defined on any given family of rooted ordered trees (defined by degree restrictions). Taking limits, we confirm, via elementary methods, the fact, due to David Aldous, and expanded by Svante Janson and others, that the limiting (scaled) distributions are all the same, and coincide with the limiting distribution of th…
Description of cocommutative Hopf algebras associated with families of trees. Applications include Cayley's theorem on the number of rooted trees with n nodes, and Catalan's theorem on the number of rooted ordered trees with n nodes.
Given integers $p\ge q>1$, a family of sets satisfies the $(p,q)$ property if among any $p$ members of it some $q$ intersect. We prove that for any fixed integer constants $p\ge q>1$, a family of $d$-intervals satisfying the $(p,q)$ property can be pierced by $O(d^{\frac{q}{q-1}})$ points, with constants depending only on $p$ and $q$. This extends results of Tardos, Kaiser and Alon for the case $q=2$, and of Kaiser and Rabinovich for the case $p=q=\lceil log_2(d+2) \rceil$. We further show that similar bounds hold in families of subgraphs of a tree or a graph of bounded tree-width, each consis…
For a tree $T$, let $i_T(t)$ be the number of independent sets of size $t$ in $T$. It is an open question, raised by Alavi, Malde, Schwenk and Erdős, whether the sequence $(i_T(t))_{t \geq 0}$ is always unimodal. Here we answer the question in the affirmative for some recursively defined families of trees, specifically paths with auxiliary trees dropped from the vertices in a periodic manner. In particular, extending a result of Wang and B.-X. Zhu, we show unimodality of the independent set sequence of a path on $2n$ vertices with $\ell_1$ and $\ell_2$ pendant edges dropped alternately from th…
The transmission of a vertex $v$ of a graph $G$ is the sum of distances from $v$ to all the other vertices in $G$. A graph is transmission irregular if all of its vertices have pairwise different transmissions. A starlike tree $T(k_1,\ldots,k_t)$ is a tree obtained by attaching to an isolated vertex $t$ pendant paths of lengths $k_1,\ldots,k_t$, respectively. It is proved that if a starlike tree $T(a,a+1,\ldots,a+k)$, $k\ge 2$, is of odd order, then it is transmission irregular. $T(1,2,\ldots,\ell)$, $\ell \ge 3$, is transmission irregular if and only if $\ell \notin \{r^2 + 1:\ r\ge 2\}$. Add…
Refining a classical proof of Whitney, we show that any $4$-connected planar triangulation can be decomposed into a Hamiltonian path and two trees. Therefore, every $4$-connected planar graph decomposes into three forests, one having maximum degree at most $2$. We use this result to show that any Hamiltonian planar triangulation can be decomposed into two trees and one spanning tree of maximum degree at most $3$. These decompositions improve the result of Gonçalves [Covering planar graphs with forests, one having bounded maximum degree. J. Comb. Theory, Ser. B, 100(6):729--739, 2010] that ever…
Let ${\cal T}=(T,w)$ be a weighted finite tree with leaves $1,..., n$.For any $I :=\{i_1,..., i_k \} \subset \{1,...,n\}$, let $D_I ({\cal T})$ be the weight of the minimal subtree of $T$ connecting $i_1,..., i_k$; the $D_{I} ({\cal T})$ are called $k$-weights of ${\cal T}$. Given a family of real numbers parametrized by the $k$-subsets of $\{1,..., n\}$, $\{D_I\}_{I \in {\{1,...,n\} \choose k}}$, we say that a weighted tree ${\cal T}=(T,w)$ with leaves $1,..., n$ realizes the family if $D_I({\cal T})=D_I$ for any $ I $. In 2006 Levy, Yoshida and Pachter defined, for any positive-weighted tree…
The Gyarfas-Sumner conjecture asserts that if H is a tree then every graph with bounded clique number and very large chromatic number contains H as an induced subgraph. This is still open, although it has been proved for a few simple families of trees, including trees of radius two, some special trees of radius three, and subdivided stars. These trees all have the property that their vertices of degree more than two are clustered quite closely together. In this paper, we prove the conjecture for two families of trees which do not have this restriction. As special cases, these families contain …
In this work we analyze bucket increasing tree families. We introduce two simple stochastic growth processes, generating random bucket increasing trees of size $n$, complementing the earlier result of Mahmoud and Smythe for bucket recursive trees. On the combinatorial side, we define multilabelled generalizations of the tree families $d$-ary increasing trees and generalized plane-oriented recursive trees. Additionally, we introduce a clustering process for ordinary increasing trees and relate it to bucket increasing trees. We discuss in detail the bucket size two and present a bijection betwee…
A family of sets is intersecting if every pair of its sets intersect. A star is a family with some element (a center) in each of its sets. The classical 1961 result of Erdős, Ko, and Rado states that every intersecting family of r-sets with $r\leq n/2$ has size at most that of a star. We say that graph G is r-EKR if, among all intersecting families of independent r-sets of G, the largest is attained by a star. In 2005 Holroyd and Talbot conjectured that every graph G is r-EKR for all $1\leq r\leq μ(G)/2$, where $μ(G)$ is the size of the smallest maximal independent set in G. We verified the co…
Bona [2007+] studied the distribution of ascents, plateaux and descents in the class of Stirling permutations, introduced by Gessel and Stanley [1978]. Recently, Janson [2008+] showed the connection between Stirling permutations and plane recursive trees and proved a joint normal law for the parameters considered by Bona. Here we will consider generalized Stirling permutations extending the earlier results of Bona and Janson, and relate them with certain families of generalized plane recursive trees, and also $(k+1)$-ary increasing trees. We also give two different bijections between certain f…
We prove that any quasirandom graph with $n$ vertices and $rn$ edges can be decomposed into $n$ copies of any fixed tree with $r$ edges. The case of decomposing a complete graph establishes a conjecture of Ringel from 1963.
We consider a dichotomy for analytic families of trees stating that either there is a colouring of the nodes for which all but finitely many levels of every tree are nonhomogeneous, or else the family contains an uncountable antichain. This dichotomy implies that every nontrivial Souslin poset satisfying the countable chain condition adds a splitting real. We then reduce the dichotomy to a conjecture of Sperner Theory. This conjecture is concerning the asymptotic behaviour of the product of the sizes of the m-shades of pairs of cross-t-intersecting families.
A vertex $v\in V$ is said to resolve two vertices $x$ and $y$ if $d_G(v,x)\ne d_G(v,y)$. A set $S\subset V$ is said to be a metric generator for $G$ if any pair of vertices of $G$ is resolved by some element of $S$. A minimum metric generator is called a metric basis, and its cardinality, $\dim(G)$, the \emph{metric dimension} of $G$. A set $S\subseteq V$ is said to be a simultaneous metric generator for a graph family ${\cal G}=\{G_1,G_2,\ldots,G_k\}$, defined on a common (labeled) vertex set, if it is a metric generator for every graph of the family. A minimum cardinality simultaneous metric…
We look at a family of meta-Fibonacci sequences which arise in studying the number of leaves at the largest level in certain infinite sequences of binary trees, restricted compositions of an integer, and binary compact codes. For this family of meta-Fibonacci sequences and two families of related sequences we derive ordinary generating functions and recurrence relations. Included in these families of sequences are several well-known sequences in the Online Encyclopedia of Integer Sequences (OEIS).
Motivated by a conjecture of Gyárfás, recently Böttcher, Hladký, Piguet, and Taraz showed that every collection $T_1,\dots,T_t$ of trees on $n$ vertices with $\sum_{i=1}^te(T_i)\leq \binom{n}{2}$ and with bounded maximum degree, can be packed into the complete graph on $(1+o(1))n$ vertices. We generalise this result where we relax the restriction of packing families of trees to families of graphs of any given non-trivial minor-closed class of graphs.
P.L. Erdos and L.A. Szekely [Adv. Appl. Math. 10(1989), 488-496] gave a bijection between rooted semilabeled trees and set partitions. L.H. Harper's results [Ann. Math. Stat. 38(1967), 410-414] on the asymptotic normality of the Stirling numbers of the second kind translates into asymptotic normality of rooted semilabeled trees with given number of vertices, when the number of internal vertices varies. The Erdos-Szekely bijection specializes to a bijection between phylogenetic trees and set partitions with classes of size \geq 2. We consider modified Stirling numbers of the second kind that en…
We present a bijective algorithm with which an arbitrary permutation decomposes canonically into elementary blocks which we call families, which are sets with a specified number of ascents and descents. We show that families, arranged in an arbitrary order in a sequence, are in bijection with permutations. The permutation decomposes canonically, by inserting parentheses, into a tree having as nodes a class of permutations which we call primitive. Primitive permutations can be assembled from very simple data. The data for the trees into which a permutation decomposes can be written in a form si…
In "Square partitions and Catalan numbers" (arXiv0912.4983), Bennett et al. presented a recursive algorithm to create a family of partitions from one or several partitions. They were mainly interested in the cases when we begin with a single square partition or with several partitions with only one part. The cardinalities of those families of partitions are the Catalan and ballot numbers, respectively. In this paper we present a closed description for those families. We also present bijections between those sets of partitions and sets of trees and forests enumerated by the Catalan an ballot nu…
Let $\mathcal G$ be a separable family of graphs. Then for all positive constants $ε$ and $Δ$ and for every sufficiently large integer $n$, every sequence $G_1,\dotsc,G_t\in\mathcal G$ of graphs of order $n$ and maximum degree at most $Δ$ such that $e(G_1)+\dotsb+e(G_t) \leq (1-ε)\binom{n}{2}$ packs into $K_n$. This improves results of Böttcher, Hladký, Piguet, and Taraz when $\mathcal G$ is the class of trees and of Messuti, Rödl, and Schacht in the case of a general separable family. The result also implies approximate versions of the Oberwolfach problem and of the Tree Packing Conjecture of…
The fractional arboricity of a digraph $D$, denoted by $γ(D)$, is defined as $γ(D)= \max_{H \subseteq D, |V(H)| >1} \frac {|A(H)|} {|V(H)|-1}$. Frank in [Covering branching, Acta Scientiarum Mathematicarum (Szeged) 41 (1979), 77-81] proved that a digraph $D$ decomposes into $k$ branchings, if and only if $Δ^{-}(D) \leq k$ and $γ(D) \leq k$. In this paper, we study digraph analogues for the Nine Dragon Tree Conjecture. We conjecture that, for positive integers $k$ and $d$, if $D$ is a digraph with $γ(D) \leq k + \frac{d-k}{d+1}$ and $Δ^{-}(D) \leq k+1$, then $D$ decomposes into $k + 1$ branchin…
We show the geometrical structure of the moduli space of positive-weighted trees with $n$ labels $1,\ldots , n$ which realize the same family of positive $(n-1)$-weights and we characterize them as a family of positive multi-weights.
André proved that the number of alternating permutations on $\{1, 2, \dots, n\}$ is equal to the Euler number $E_n$. A refinement of André's result was given by Entringer, who proved that counting alternating permutations according to the first element gives rise to Seidel's triangle $(E_{n,k})$ for computing the Euler numbers. In a series of papers, using generating function method and induction, Poupard gave several further combinatorial interpretations for $E_{n,k}$ both in alternating permutations and increasing trees. Kuznetsov, Pak, and Postnikov have given more combinatorial interpretat…
The theory of sparse structures usually uses tree like structures as building blocks. In the context of sparse/dense dichotomy this role is played by graphs with bounded tree depth. In this paper we survey results related to this concept and particularly explain how these graphs are used to decompose and construct more complex graphs and structures. In more technical terms we survey some of the properties and applications of low tree depth decomposition of graphs.
We are interested in the quantitative analysis of the compaction ratio for two classical families of trees: recursive trees and plane binary increasing trees. These families are typical representatives of tree models with a small depth. Once a tree of size $n$ is compacted by keeping only one occurrence of all fringe subtrees appearing in the tree the resulting graph contains only $O(n / \ln n)$ nodes. This result must be compared to classical results of compaction in the families of simply generated trees, where the analogous result states that the compacted structure is of size of order $n /…
In this paper, we introduce the notion of the quadratic graph, that is a graph whose eigenvalues are integral or quadratic algebraic integral, and determine nine infinite families of quadratic starlike trees, which are just all the quadratic starlike trees including integral starlike trees. Thus the quadratic starlike trees are completely characterized, and moreover, the display expressions for the characteristic polynomials of the quadratic starlike trees are also given.
We apply a tree-based methodology to solve new, very broadly defined families of nested recursions of the general form R(n)=sum_{i=1}^k R(n-a_i-sum_{j=1}^p R(n-b_{ij})), where a_i are integers, b_{ij} are natural numbers, and k,p are natural numbers that we use to denote "arity" and "order," respectively, and with some specified initial conditions. The key idea of the tree-based solution method is to associate such recursions with infinite labelled trees in a natural way so that the solution to the recursions solves a counting question relating to the corresponding trees. We characterize certa…
We asymptotically determine the size of the largest family F of subsets of {1,...,n} not containing a given poset P if the Hasse diagram of P is a tree. This is a qualitative generalization of several known results including Sperner's theorem.
Given an edge-weighted tree $T$ with $n$ leaves, sample the leaves uniformly at random without replacement and let $W_k$, $2 \le k \le n$, be the length of the subtree spanned by the first $k$ leaves. We consider the question, "Can $T$ be identified (up to isomorphism) by the joint probability distribution of the random vector $(W_2, \ldots, W_n)$?" We show that if $T$ is known {\em a priori} to belong to one of various families of edge-weighted trees, then the answer is, "Yes." These families include the edge-weighted trees with edge-weights in general position, the ultrametric edge-weighted …
We investigate clique trees of infinite locally finite chordal graphs. Our main contribution is a bijection between the set of clique trees and the product of local finite families of finite trees. Even more, the edges of a clique tree are in bijection with the edges of the corresponding collection of finite trees. This allows us to enumerate the clique trees of a chordal graph and extend various classic characterisations of clique trees to the infinite setting.
The Strong Nine Dragon Tree Conjecture asserts that for any integers $k$ and $d$ any graph with fractional arboricity at most $k + \frac{d}{d+k+1}$ decomposes into $k+1$ forests, such that for at least one of the forests, every connected component contains at most $d$ edges. We prove this conjecture when $d \leq k+1$. We also prove an approximate version of this conjecture, that is, we prove that for any positive integers $k$ and $d$, any graph with fractional arboricity at most $k + \frac{d}{d+k+1}$ decomposes into $k+1$ forests, such that one for at least one of the forests, every connected …
We consider uniform random permutations drawn from a family enumerated through generating trees. We develop a new general technique to establish a central limit theorem for the number of consecutive occurrences of a fixed pattern in such permutations. We propose a technique to sample uniform permutations in such families as conditioned random colored walks. Building on that, we derive the behavior of the consecutive patterns in random permutations studying properties of the consecutive increments in the corresponding random walks. The method applies to families of permutations with a one-dimen…
The local tree-width of a graph G=(V,E) is the function ltw^G: N -> N that associates with every natural number r the maximal tree-width of an r-neighborhood in G. Our main graph theoretic result is a decomposition theorem for graphs with excluded minors that essentially says that such graphs can be decomposed into trees of graphs of bounded local tree-width. As an application of this theorem, we show that a number of combinatorial optimization problems, such as Minimum Vertex Cover, Minimum Dominating Set, and Maximum Independent Set have a polynomial time approximation scheme when restricted…
We prove that for any positive integers $k$ and $d$, if a graph $G$ has maximum average degree at most $2k + \frac{2d}{d+k+1}$, then $G$ decomposes into $k+1$ pseudoforests $C_{1},\ldots,C_{k+1}$ such that there is an $i$ such that for every connected component $C$ of $C_{i}$, we have that $e(C) \leq d$.
Let $G$ be an $n$-vertex connected graph. A cyclic base ordering of $G$ is a cyclic ordering of all edges such that every cyclically consecutive $n-1$ edges induce a spanning tree of $G$. In this project, we study cyclic base ordering of triangular grid graphs, series composition of graphs, generalized theta graphs, and circulant graphs.
In this paper, we first prove that if the edges of $K_{2m}$ are properly colored by $2m-1$ colors in such a way that any two colors induce a 2-factor of which each component is a 4-cycle, then $K_{2m}$ can be decomposed into $m$ isomorphic multicolored spanning trees. Consequently, we show that there exist three disjoint isomorphic multicolored spanning trees in any properly (2$m-$1)-edge-colored $K_{2m}$ for $m\geq 14$.
Let ${\cal T}=(T,w)$ be a weighted finite tree with leaves $1,..., n$.For any $I :=\{i_1,..., i_k \} \subset \{1,...,n\}$,let $D_I ({\cal T})$ be the weight of the minimal subtree of $T$ connecting $i_1,..., i_k$; the $D_{I} ({\cal T})$ are called $k$-weights of ${\cal T}$. Given a family of real numbers parametrized by the $k$-subsets of $ \{1,..., n\}$, $\{D_I\}_{I \in {\{1,...,n\} \choose k}}$, we say that a weighted tree ${\cal T}=(T,w)$ with leaves $1,..., n$ realizes the family if $D_I({\cal T})=D_I$ for any $ I $. In [P-S] Pachter and Speyer proved that, if $3 \leq k \leq (n+1)/2$ and $…
For each poset $H$ whose Hasse diagram is a tree of height $k$, we show that the largest size of a family $\cF$ of subsets of $[n]=\{1,..., n\}$ not containing $H$ as an induced subposet is asymptotic to $(k-1){n\choose \fl{n/2}}$. This extends the result of Bukh \cite{bukh}, which in turn generalizes several known results including Sperner's theorem.
Combining tree decomposition and transfer matrix techniques provides a very general algorithm for computing exact partition functions of statistical models defined on arbitrary graphs. The algorithm is particularly efficient in the case of planar graphs. We illustrate it by computing the Potts model partition functions and chromatic polynomials (the number of proper vertex colourings using Q colours) for large samples of random planar graphs with up to N=100 vertices. In the latter case, our algorithm yields a sub-exponential average running time of ~ exp(1.516 sqrt(N)), a substantial improvem…
In this paper, we focus on families of bipartitions, i.e. set partitions consisting of at most two components. We say that a family of bipartitions is a separating family for a set $S$ if every two elements in $S$ can be separated by some bipartition. Furthermore, we call a separating family minimal if no proper subfamily is a separating family. We characterize the set of all minimal separating families of maximum size for arbitrary set $S$ as the set of all spanning trees on $S$ and enumerate minimal separating families of maximum size. Furthermore, we enumerate separating families of arbitra…
