Related papers: Caterpillars and alternating paths

We prove that any convex body in the plane can be partitioned into $m$ convex parts of equal areas and perimeters for any integer $m\ge 2$; this result was previously known for prime powers $m=p^k$. We also discuss possible…

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…

We study codes with parameters of $q$-ary shortened Hamming codes, i.e., $(n=(q^m-q)/(q-1), q^{n-m}, 3)_q$. Firstly, we prove the fact mentioned in 1998 by Brouwer et al. that such codes are optimal, generalizing it to a bound for multifold…

Fix any two numbers $p$ and $q$, with $1<p<q$; we give an example of an integral functional enjoying uniform ellipticity and $p$-$q$ growth.

In this paper, we investigate the structures of an extremal tree which has the minimal number of subtrees in the set of all trees with the given degree sequence of a tree. In particular, the extremal trees must be caterpillar and but in…

Denote by $p_m$ the $m$-th prime number ($p_1=2,~p_2=3,~p_3=5,~ p_4=7,~\ldots$). Let $T$ be a rooted tree with branches $T_1,T_2,\ldots,T_r$. The Matula number $M(T)$ of $T$ is $p_{M(T_1)}\cdot p_{M(T_2)}\cdot \ldots \cdot p_{M(T_r)}$,…

We solve a recent question of Caro, Patk\'os and Tuza by determining the exact maximum number of edges in a bipartite connected graph as a function of the longest path it contains as a subgraph and of the number of vertices in each side of…

We investigate the structure of trees that have minimal algebraic connectivity among all trees with a given degree sequence. We show that such trees are caterpillars and that the vertex degrees are non-decreasing on every path on…

We consider Bernoulli bond percolation on oriented regular trees, where besides the usual short bonds, all bonds of a certain length are added. Independently, short bonds are open with probability $p$ and long bonds are open with…

Let $\Pi_q$ be an arbitrary finite projective plane of order $q$. A subset $S$ of its points is called saturating if any point outside $S$ is collinear with a pair of points from $S$. Applying probabilistic tools we improve the upper bound…

In this paper, we estimate the weak saturation numbers of trees. As a case study, we examine caterpillars and obtain several tight estimates. In particular, this implies that for any $\alpha\in [1,2]$, there exist caterpillars with $k$…

Untangling is a process in which some vertices of a planar graph are moved to obtain a straight-line plane drawing. The aim is to move as few vertices as possible. We present an algorithm that untangles the cycle graph C_n while keeping at…

Let P be a set of n weighted points, Q be a set of m unweighted points in the plane, and k a non-negative integer. We consider the problem of computing a subset $Q'\subseteq Q$ with size at most k such that the sum of the weights of the…

We consider the following question: How many edge-disjoint plane spanning trees are contained in a complete geometric graph $GK_n$ on any set $S$ of $n$ points in general position in the plane? We show that this number is in…

A drawing of a graph is $k$-plane if every edge contains at most $k$ crossings. A $k$-plane drawing is saturated if we cannot add any edge so that the drawing remains $k$-plane. It is well-known that saturated $0$-plane drawings, that is,…

Let the columns of a $p \times q$ matrix $M$ over any ring be partitioned into $n$ blocks, $M = [M_1, ..., M_n]$. If no $p \times p$ submatrix of $M$ with columns from distinct blocks $M_i$ is invertible, then there is an invertible $p…

We give exact values and bounds on the isoperimetric peak of complete trees, improving on known results. For the complete $q$-ary tree of depth $d$, if $q\ge 5$, then we find that the isoperimetric peak equals $d$, completing an open…

Tree sets are abstract structures that can be used to model various tree-shaped objects in combinatorics. Finite tree sets can be represented by finite graph-theoretical trees. We extend this representation theory to infinite tree sets.…

We present a new definition of non-ambiguous trees (NATs) as labelled binary trees. We thus get a differential equation whose solution can be described combinatorially. This yield a new formula for the number of NATs. We also obtain…

Two distinct systems of commutative complex numbers in n dimensions are described, of polar and planar types. Exponential forms of n-complex numbers are given in each case, which depend on geometric variables. Azimuthal angles, which are…