Related papers: Klazar trees and perfect matchings
We give constructions to realize an odd number, which is representable as sum of two squares, as determinant of an achiral knot, thus proving that these are exactly the numbers occurring as such determinants. Later we study which numbers…
In this paper we prove a combinatorial theorem for finite labellings of trees, and show that it is equivalent to a theorem for finite covers of metric trees and a fixed point theorem on metric trees. We trace how these connections mimic the…
We introduce a new metric of match, called Cartesian tree matching, which means that two strings match if they have the same Cartesian trees. Based on Cartesian tree matching, we define single pattern matching for a text of length n and a…
This work addresses an enumeration problem on weighted bi-colored plane trees with prescribed vertex data, with all vertices labeled distinctly. We give a bijection proof of the enumeration formula originally due to Kochetkov, hence…
A subset of vertices is a {\it maximum independent set} if no two of the vertices are adjacent and the subset has maximum cardinality. A subset of vertices is called a {\it maximum dissociation set} if it induces a subgraph with vertex…
A conjecture by Bollob\'as from 1995 (which is a weakenning of the famous Tree Packing Conjecture by Gy\'arf\'as from 1976) states that any set of $k$ trees $T_n,T_{n-1},\dots,T_{n-k+1}$, such that $T_{n-i}$ has $n-i$ vertices, pack into…
Weighted recursive trees are built by adding successively vertices with predetermined weights to a tree: each new vertex is attached to a parent chosen randomly proportionally to its weight. Under some assumptions on the sequence of…
We represent the generalized Collatz function with the recursive ruler function r(2n) = r(n) + 1 and r(2n + 1) = 1. We generate even-only and odd-only Collatz subsequences that contain significantly fewer elements term by term, to 2 and 1,…
A widely used method for determining the similarity of two labeled trees is to compute a maximum agreement subtree of the two trees. Previous work on this similarity measure is only concerned with the comparison of labeled trees of two…
Cartesian tree pattern matching consists of finding all the factors of a text that have the same Cartesian tree than a given pattern. There already exist theoretical and practical solutions for the exact case. In this paper, we propose the…
The Wiener index is maximized over the set of trees with the given vertex weight and degree sequences. This model covers the traditional "unweighed" Wiener index, the terminal Wiener index, and the vertex distance index. It is shown that…
The famous tree packing conjecture of Gy\'arf\'as from 1976 says that any sequence of trees $T_1,\ldots,T_n$ such that $|T_i|=i$ for each $i\in [n]$ packs into the complete $n$-vertex graph $K_n$. Packing even just the largest trees in such…
Pairwise ordered tree alignment are combinatorial objects that appear in RNA secondary structure comparison. However, the usual representation of tree alignments as supertrees is ambiguous, i.e. two distinct supertrees may induce identical…
The classical matrix tree theorem relates the number of spanning trees of a connected graph with the product of the nonzero eigenvalues of its Laplacian matrix. The class of regular matroids generalizes that of graphical matroids, and a…
Over some types of trees with a given number of vertices, which trees minimize or maximize the total number of subtrees or leaf containing subtrees are studied. Here are some of the main results:\ (1)\, Sharp upper bound on the total number…
The 1970s conjecture of Lov\'asz and Plummer that the number of perfect matchings in any $3$-regular graph is exponential in the number of vertices was proved in 2011 by Esperet, Kardo\v{s}, King, Kr\'al', and Norine. We give the exact…
Sorting is a foundational problem in computer science that is typically employed on sequences or total orders. More recently, a more general form of sorting on partially ordered sets (or posets), where some pairs of elements are…
We consider the problem of constructing an an optimal-weight tree from the 3*(n choose 4) weighted quartet topologies on n objects, where optimality means that the summed weight of the embedded quartet topologiesis optimal (so it can be the…
We consider combinatorial aspects of $\lambda$-terms in the model based on de Bruijn indices where each building constructor is of size one. Surprisingly, the counting sequence for $\lambda$-terms corresponds also to two families of binary…
We present a version of the weighted cellular matrix-tree theorem that is suitable for calculating explicit generating functions for spanning trees of highly structured families of simplicial and cell complexes. We apply the result to give…