Related papers: Putting Dots in Triangles
We study the maximal number of pairwise distinct columns in a $\Delta$-modular integer matrix with $m$ rows. Recent results by Lee et al. provide an asymptotically tight upper bound of $O(m^2)$ for fixed $\Delta$. We complement this and…
A linear chord diagram of size $n$ is a partition of the set $\{1,2,\cdots,2n\}$ into sets of size two, called chords. From a table showing the number of linear chord diagrams of degree $n$ such that every chord has length at least $k$, we…
This paper attacks the following problem. We are given a large number $N$ of rectangles in the plane, each with horizontal and vertical sides, and also a number $r<N$. The given list of $N$ rectangles may contain duplicates. The problem is…
We prove that, among rectangular grid graphs with a fixed number of vertices, the number of spanning trees increases when the side lengths are made more balanced. In particular, among all rectangular grid graphs with $n^2$ vertices, the…
We prove that almost every triangle can be dissected only into $n^2$ triangles which have to be equal one another. Moreover, such a dissection is unique for every $n$. It turns out that to solve this "simple" problem it is convenient to use…
In 1975, P. Erd\"{o}s proposed the problem of determining the maximum number $f(n)$ of edges in a graph of $n$ vertices in which any two cycles are of different lengths. In this paper, it is proved that $$f(n)\geq n+32t-1$$ for…
We answer an open problem in the \emph{American Mathematical Monthly} about covering large triangles. Given a triangle $T$ of any triangular shape with a selected side length between $n \in \mathbb{N}$ and $n+1$, Baek and Lee proved that…
What is the maximum number of holes that a polyomino with $n$ tiles can enclose? Call this number $f(n)$. We show that if $n_k = \left( 2^{2k+1} + 3 \cdot 2^{k+1}+4 \right) / 3$ and $h_k = \left( 2^{2k}-1 \right) /3$, then $f(n_k) = h_k$…
We present a formula for the number of line segments connecting q+1 points of an n_1 x...x n_k rectangular grid. As corollaries, we obtain formulas for the number of lines through at least q points and, respectively, through exactly q…
The celebrated Mantel's theorem states that any triangle-free graph on $n$ vertices contains at most $\left\lfloor n^2/4\right\rfloor$ edges. It is natural to ask how many triangles must exist in a graph with more than $\left\lfloor…
A graph whose vertices are points in the plane and whose edges are noncrossing straight-line segments of unit length is called a \emph{matchstick graph}. We prove two somewhat counterintuitive results concerning the maximum number of edges…
An edge coloring of the n-vertex complete graph K_n is a Gallai coloring if it does not contain any rainbow triangle, that is, a triangle whose edges are colored with three distinct colors. We prove that the number of Gallai colorings of…
Let $G$ be an edge-colored graph on $n$ vertices. The minimum color degree of $G$, denoted by $\delta^c(G)$, is defined as the minimum number of colors assigned to the edges incident to a vertex in $G$. In 2013, H. Li proved that an…
Given $n$ lines in general position in the plane, how many bounded triangular faces can the arrangement have? We construct a straight-line affine arrangement of $19$ lines satisfying the conditions of the iterative construction by…
We prove that the maximum number of edges in a 3-uniform linear hypergraph on $n$ vertices containing no 2-regular subhypergraph is $n^{1+o(1)}$. This resolves a conjecture of Dellamonica, Haxell, Luczak, Mubayi, Nagle, Person, R\"odl,…
Let $D$ be a knot diagram, and let ${\mathcal D}$ denote the set of diagrams that can be obtained from $D$ by crossing exchanges. If $D$ has $n$ crossings, then ${\mathcal D}$ consists of $2^n$ diagrams. A folklore argument shows that at…
We show that any combinatorial triangulation on n vertices can be transformed into a 4-connected one using at most floor((3n - 9)/5) edge flips. We also give an example of an infinite family of triangulations that requires this many flips…
How can you fill a $3\times 3$ grid with the letters A and M so that the word ``AMM'' appears as many times as possible in the grid? More generally, given a word $w$ of length $n$, how can you fill an $n\times n$ grid so that $w$ appears as…
We show that each set of $n\ge 2$ points in the plane in general position has a straight-line matching with at least $(5n+1)/27$ edges whose segments form a connected set, and such a matching can be computed in $O(n \log n)$ time. As an…
We find the maximal number of regions that a straight line embedding of a N-cycle graph can enclose.