2
$\begingroup$

A bicolored graph is a graph each vertex of which has been assigned one of two colors such that each edge connects vertices of different colors. A bipartite graph is a graph $G$ which admits such a coloring. Given $j$ white and $i$ black vertices, there are $2^{ji}$ ways to join vertices of different colors. Thus the number of (labeled) bicolored graphs on n vertices is $$b_n=\sum_{i+j=n}\binom{n}i2^{ij}.$$

Question. Is there a combinatorial proof for this identity? $$\sum_{i+j=n}\binom{n}{i,j}2^{ij}=2\sum_{i+j+\ell=n-1}\binom{n-1}{i,j,\ell}2^{ij+j\ell}.$$

$\endgroup$

1 Answer 1

3
$\begingroup$

RHS also enumerates bipartite labelled graphs. Fix a vertex $v$. We choose a color for it (multiple 2 in RHS), then we choose $j$ vertices of the same color and $i$ neighbors of $v$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.