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We give here a review of results about double bialgebras, that is to saybialgebras with two coproducts, the first one being a comodule morphism for thecoaction induced by the second one. An accent is put on the case of connectedbialgebras. The subjects of these results are the monoid of characters andtheir actions, polynomial invariants, the antipode and the eulerian idempotent.As examples, they are applied on a double bialgebra of graphs and onquasishuffle bialgebras. This includes a new proof of a combinatorialinterpretation of the coefficients of the chromatic polynomial due to Greeneand Zaslavsky.

Bialgebras in cointeraction, the antipode and the eulerian idempotent