Prove that if a graph has an Eulerian path, then the number of odd ...
Oct 14, 2023 · Now, let's use these properties to prove the statement. If a graph has an Eulerian path, there must be exactly two vertices with odd degrees (the starting and ending vertices) …
Graph theory: adjacency vs incident - Mathematics Stack Exchange
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 …
geometry - Orientation of a triangle's vertices in 3D space: …
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 …
Proving that the number of vertices of odd degree in any graph G …
I'm having a bit of a trouble with the below question Given G is an undirected graph, the degree of a vertex v, denoted by deg(v), in graph G is the number of neighbors of v. Prove that the …
Show that a connected graph on $n$ vertices is a tree if and only …
Here's an 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:
polyhedra - Polyhedron with least number of vertices whose …
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 …
Is there a $ (3,3)$-windmill graph with $19$ vertices?
Dec 27, 2025 · The above construction provides an explicit example of a $6$ -regular graph on $19$ vertices that is locally a $ (3,3)$ -windmill. If one wishes to analyze the graph by hand …
geometry - How many verticies, edges and faces (cells) does an nd ...
Mar 19, 2021 · Here are a couple hints. I will consider the vertices to be length- n n bitstrings. An edge connects two vertices if the vertices differ by a single bit flip. How many edges are …
Orientation of a geometric simplex - ordering of its vertices
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 …
combinatorics - Every $k$ vertices in an $k$ - connected graph are ...
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 …