If the Handshaking Theorem holds for a sequence of vertices, does there always exist a graph?
18383
List of ways to tell if degree sequence is impossible for a simple graph
13226
If $n$ is a natural number $\ge 2$ how do I prove that any graph with $n$ vertices has at least two vertices of the same degree?
139657
Prove that if $\chi(G-u-v)=\chi(G)-2$ for every vertices $u, v$ ($u \ne v$) then G is complete graph.
150508
∗∗Prove that each graph with an even number of vertices has two vertices with an even number of common neighbors.
91134
In any finite graph with at least two vertices, there must be two vertices with the same degree
27661
The closed graph theorem in topology
39753
Proving that in a simple graph with $n$ vertices, two vertices have the same degree
17868
How to calculate the number of possible connected simple graphs with $n$ labelled vertices
122691
Graphs of Linear Inequalities
23916
Does 3-partite graph with at least n+1 edges per vertex have a triangle?
1453
Prove that the graph is connected
33822
Is there a monoid structure on the set of paths of a graph?
53924
What is the smallest $n$ for which loopless graphs contain non-isomorphic $n$-vertex graphs with the same degree sequence?
126428
Prove the edges of a multigraph may be oriented such that the net-degree of any vertex is $\leq 1$.
49193
Constructing a graph from a degree sequence
57954
How many transitive relations on a set of $n$ elements?
19807
Connected graph with colored edges