If every vertex of a planar graph has an even degree, it’s 2-colourable
37984
Prove a planar graph with all regions having even degree can be colored with 2 colors
14932
What would be the number of inequivalent $6$-colourings of the faces of a cube?
141482
Proof that plane with $N$ lines can be painted with two colors so that any two neighboring regions are painted in different colors
35092
Match covered graph is 2-connected
48692
A problem about the graph theory
126428
Prove the edges of a multigraph may be oriented such that the net-degree of any vertex is $\leq 1$.
49585
Proof a graph is bipartite if and only if it contains no odd cycles
38700
the Nordhaus-Gaddum problems for chromatic number of graph and its complement
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 $1$
57930
Chromatic polynomial properties
150508
∗∗Prove that each graph with an even number of vertices has two vertices with an even number of common neighbors.
19807
Connected graph with colored edges
91134
In any finite graph with at least two vertices, there must be two vertices with the same degree
52261
Proof that any self-complementary graph has to have $4k$ or $4k+1$ vertex, for some $k \in \mathbb N$
96620
Condition on degrees for existence of a tree
72417
Bipartite graphs
24477
Monochromatic squares in a colored plane