Hi i didnt find good information on the web about bipartite graps.
For example does the sum of the degrees on both sides have to be equal? or a bipartite graph G (with side A & B) whose number of edges is 27 can have sum(deg(A)) = 24? or it must be 27 too? (so we will get that 2|E|=54)
How can i check if it has no odd circles? never seen a formula for this
Thanks in advance
A graph $G\equiv\left(V,E\right)$ is bipartite if there exist disjoint sets $A,B$ satisfying $V=A\cup B$ s.t. for all $a,a^{\prime}\in A$, $\left(a,a^{\prime}\right)\notin E$ and for all $b,b^{\prime}\in B$, $\left(b,b^{\prime}\right)\notin E$.
From this it follows that outdeg$\left(A\right)$ $=$ indeg$\left(B\right)$ and vice-versa (convince yourself of this).
A graph $G$ is bipartite if and only if it does not contain an odd cycle. Suppose a graph contains an odd cycle. Since there is no way to make a bipartite subgraph out of the odd cycle (convince yourself of this), $G$ is not bipartite. The proof of the converse is slightly more involved: https://proofwiki.org/wiki/Graph_is_Bipartite_iff_No_Odd_Cycles