Related papers: No triangle can be cut into seven congruent triang…
A classical result of von Staudt states that if eight planes osculate a twisted cubic curve and we divide them into two groups of four, then the eight vertices of the corresponding tetrahedra lie on a twisted cubic curve. In the current…
We study graphs where each edge adjacent to a vertex of small degree (7 and 9, respectively) belongs to many triangles (4 and 5, respectively) and show that these graphs contain a complete graph (K_6 and K_7, respectively) as a minor. The…
It is conjectured that every integer N>454 is the sum of seven nonnegative cubes. We prove the conjecture when N is congruent to 2 mod 4. This result, together with a recent proof for 4|N, shows that the conjecture is true for all even N.
A point in the interior of a planar triangle determines a subdivision into six subtriangles. A triangle with angles commensurable with $\pi$ is called $\pi$-commensurable. For such a triangle a subdivision where each of the subtriangles are…
We prove that any element in a matroid can be removed, by either deletion or contraction, in such a way that no tangle "splits".
We show that if all the eigenvalues of a tree are integers, then it does not contain a subdivided edge with 7 vertices.
We introduce and study a new kind of congruent number problem on the right trapezoid.
We show geometrically that $\sqrt n$ is irrational for $n=3,5,7$ by adapting Tennenbaum's geometric proof that $\sqrt 2$ is irrational. We also show that this method cannot be used to prove the irrationality of $\sqrt n$ for a bigger $n$.
We prove a discrete analog of a certain four-vertex theorem for space curves. The smooth case goes back to the work of Beniamino Segre and states that a closed and smooth curve whose tangent indicatrix has no self-intersections admits at…
A simpler proof of the four color theorem is presented. The proof was reached using a series of equivalent theorems. First the maximum number of edges of a planar graph is obatined as well as the minimum number of edges for a complete…
We prove that a family $\mathcal{T}$ of distinct triangles on $n$ given vertices that does not have a rainbow triangle (that is, three edges, each taken from a different triangle in $\mathcal{T}$, that form together a triangle) must be of…
In 2007, Ando and Egawa proved a theorem which provides a lower bound on the number of contractible edges preserving $4$-connectedness in $4$-connected graphs. In this paper, we refine their bounds, especially for the $4$-connected plane…
We show that if a coloring of the plane has the properties that any two points at distance one are colored differently and the plane is partitioned into uniformly colored triangles under certain conditions, then it requires at least seven…
We study the problem of construction of a triangle from the feet of its internal angle bisectors. It is proved that in general case ruler-and-compass solution of this problem is impossible.
We solve a problem of Dujmovi\'c and Wood (2007) by showing that a complete convex geometric graph on $n$ vertices cannot be decomposed into fewer than $n-1$ star-forests, each consisting of noncrossing edges. This bound is clearly tight.…
Motivated by Chudnovsky's structure theorem of bull-free graphs, Abu-Khzam, Feghali, and M\"uller have recently proved that deciding if a graph has a vertex partition into disjoint cliques and a triangle-free graph is NP-complete for five…
A triple system is cancellative if no three of its distinct edges satisfy $A \cup B=A \cup C$. It is tripartite if it has a vertex partition into three parts such that every edge has exactly one point in each part. It is easy to see that…
Let $G$ be a simple graph and $v$ be a vertex of $G$. The triangle-degree of $v$ in $G$ is the number of triangles that contain $v$. While every graph has at least two vertices with the same degree, there are graphs in which every vertex…
We prove that deciding whether the edge set of a graph can be partitionned into two spanning trees with orientation constraints is NP-complete. If P $\neq$ NP then this disproves a conjecture of Recski.
One unsolved mathematical problem remains the perfect cuboid problem. A perfect cuboid is a rectangular parallelepiped whose edges, face diagonals and space diagonal are all expressed as integers. No such cuboid has yet been discovered and…