Second extra differential on odd graph complexes

A suitable extra differential on graph complexes can lead to a pairing of its cohomological classes. Many such extra differentials are known for various graph complexes, including Kontsevich’s graph complex $G{C}_n$ for odd $n$. In this paper we introduce another extra differential on the same graph complex, leading to another way of pairing of its cohomological classes. Two ways of pairing lead to even further understanding of graph cohomology through “waterfall mechanism”.