If there is no cycle through $e$ and $g$, then all paths go through the same vertex
120617
No cycle containing edges $e$ and $g$ implies there is a vertex $u$ so that every path sharing one end with $e$ and another with $g$ contains $u$
44673
$M$ connected $\iff$ $M$ and $\emptyset$ are the only subsets of $M$ open and closed at the same time
33822
Is there a monoid structure on the set of paths of a graph?
149609
Conditions on cycle types for permutations to generate $S_n$
72267
Prove there's a simple path of length $k$ in a simple graph $G$ where all the vertices have degree of at least $k$
129752
How to find whether it is possible for each vertex of a graph to have a different degree?
96781
connected manifolds are path connected
33316
Show that if $G$ is simple a graph with $n$ vertices and 􏰈the number of edges $m>\binom{n-1}{2}$, then $G$ is connected.
15865
Show that $X = \{ (x,y) \in\mathbb{R}^2\mid x \in \mathbb{Q}\text{ or }y \in \mathbb{Q}\}$ is path connected.
133270
Prove that the chromatic polynomial of a cycle graph $C_{n}$ equals $(k-1)^{n} + (k-1)(-1)^{n}$
101520
Let $G$ be a graph of minimum degree $k>1$. Show that $G$ has a cycle of length at least $k+1$
91253
Proving that every graph that each vertex at degree 2 must contain a cycle
110422
Prove that a graph $G$ is a forest if and only if every induced subgraph of $G$ contain a vertex of degree at most $1$
94928
Every $k$ vertices in an $k$ - connected graph are contained in a cycle.
59831
How to relate combinatorial proof to bipartite graphs?
63691
G acts on X transitively, then there exists some element that does not have any fixed points
82578
Show that if $G$ is $2-$connected, then for any two vertices $u$ and $v$ there exists a cycle $C$ such that $u, v \in V (C)$