To prove that, a graph $G$ is bi-colored iff, when in it doesn't have a cycle of odd lengths.
42513
Graph with Cycle and Two-Colorable
28877
Graph with $k$ components.
128834
Proving the number of even and odd permutations of a subgroup $H<S_{n}$ are equal, provided $H$ is not contained in $A_{n}$
20544
Proving that if $G$ has a Hamiltonian circuit, then the line graph $L(G)$ also has a Hamiltonian Circuit
141482
Proof that plane with $N$ lines can be painted with two colors so that any two neighboring regions are painted in different colors
38700
the Nordhaus-Gaddum problems for chromatic number of graph and its complement
90618
Color the edges of $K_6$ red or blue. Prove that there is a cycle of length 4 with monochromatic edges.
33216
Prove that the order of an element in $S_n$ equals the least common multiple of the lengths of the cycles in its cycle decomposition.
26449
$|G|=24$ prove that $G$ is not simple
117992
Help with graph induction proof
12557
Coloring Graph Problem
138465
Graph Coloring and Complete Graph
99781
Proof that any simple connected graph has at least 2 non-cut vertices.
117707
d-regular bipartite graph
89531
Show that every graph $G$ has a bipartite subgraph with at least half of the edges of $G$
131137
If ${\lvert{E(G)}\rvert}>{\lvert{V(G)}\rvert}+3$, graph $G$ has at least two disjoint cycles.
11329
Proving a graph has no Hamiltonian cycle