Oct 23, 2022 · I would approach the issue from a completely different direction. Consider a triangle in 3D with vertices at $\vec {v}_0$, $\vec {v}_1$, and $\vec {v}_2$. It has a directed …

Dec 6, 2025 · The least number of vertices that a polyhedron can have, such that its diagonal faces enclose an interior solid region? Note: "interior" means the solid does not intersect the …

Usually one speaks of adjacent vertices, but of incident edges. Two vertices are called adjacent if they are connected by an edge. Two edges are called incident, if they share a vertex. Also, a …

Jan 23, 2015 · To prove that the number of odd vertices in a simple graph is always even, we can use the Handshaking Lemma, which states that the sum of the degrees of all vertices in a …

I have tried some ways - mainly using induction by removing one of the vertices of the set from the graph, and/or using Menger's theorem to construct the cycle. But I always encounter …

Here's alternative proof that a connected graph with n vertices and n-1 edges must be a tree modified from yours but without having to rely on the first derivation:

Jan 14, 2022 · 2 A simple graph $G$ with $n$ vertices in which the sum of degrees of every two non-adjacent vertices is at least $n-1$ has a Hamiltonian path.

Nov 8, 2023 · My question is, what is an ordering of a simplex? Is it just a permutation of the vertices or does it have to satisfy some other rules? If it's defined to be a permutation of …

A directed simple graph is a structure consisting of the set of vertices and a binary relation that is irreflexive. For the case of the disconnected graph, the relation is empty, and there is one such …

Aug 18, 2021 · In general having three vertices of a parallelogram allows for three possible positions of the fourth vertex, depending on which pair of the three are endpoints of a diagonal …