Bipartite Graphs

$\mathrm{Let}G=(V,E)$ be a graph. A bipartition of $G$ is an ordered pair $(A,B)$ of subsets of $V$ with the properties:

• $A\cup B=V$ and $A\cap B=\mathrm{\varnothing }$
• For every $e\in E,e\cap A\ne \mathrm{\varnothing }$ and $e\cap B\ne \mathrm{\varnothing }$
• i.e. splitting the graph in a way where all edge connections are severed
• if $G$ is bipartite then every subgraph of $G$ is bipartite
  Between graphs
• if $G\simeq H,$ then $G$ is bipartite $⟺H$ is bipartite
  With Cycles
• $C_n$ is bipartite $⟺$ n is even, for $n\ge 3$
• if $G$ contains an odd cycle, then $G$ is not bipartite