Let $T_1, \: \ldots, \: T_k$ subtrees of a tree $T$ such that for all $1\leq i <j \leq k $ the trees $T_i$ and $T_j$ have a vertex in common.
23008
For all $1 \leq i < j \leq k$, the subtrees $T_i$ and $T_j$ have a vertex in common. Show that $T$ has a vertex which is in all of the $T_i$.
138643
Prove that at most $2k+1$ independent sets form the set of vertices of a graph, where $k$ is the maximum degree.
146204
Subsets sharing at most $i$ elements
142917
Graph vertex chromatic number in a union of 2 sub-graphs
27407
every planar graph can be expressed as a union of five edge-disjoint forests
96620
Condition on degrees for existence of a tree
500
Draw a graph with a common vertex
106927
How do I prove $[G:H\cap K]\leq [G:H][G:K]$?
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 $1$
67887
Prove that a graph on $n$ vertices with girth $g$ has at most $\frac{g}{g-2}(n-2)$ edges
6429
prove that a connected graph with $n$ vertices has at least $n-1$ edges
39100
Exercise from Serre's "Trees" - prove that a given group is trivial
158355
If $A$ commutes with $B$ and $B$ commutes with $C$, then $A$ commutes with $C$
158083
If $G$ is a connected graph with $n$ vertices and $n - 1$ edges then $G$ is a tree, using Induction.
129845
Prove that $T$ is a subgraph of $G$
121130
Graph with even vertices and ${n-1}\choose 2$ $+ 1$ edges has a perfect matching
131137
If ${\lvert{E(G)}\rvert}>{\lvert{V(G)}\rvert}+3$, graph $G$ has at least two disjoint cycles.