We introduce and start the study of a bialgebra of graphs,
which we call the {\it 4-bialgebra}, and of the dual {\it
bialgebra of 4-invariants}. The 4-bialgebra is
similar to the ring of graphs introduced by W.~T.~Tutte in~1946,
but its structure is more complicated.  The roots of the
definition are in low dimensional topology, namely, in
the recent theory of Vassiliev knot invariants.
In particular, 4-invariants of graphs determine
Vassiliev invariants of knots. The relation between the
two notions is discussed.