  Oriented ribbon graphs (dessins d'enfant) are graphs embedded in oriented
surfaces. A quasi-tree of a ribbon graph is a spanning subgraph with one face,
which is described by an ordered chord diagram. We show that for any link
diagram $L$, there is an associated ribbon graph whose quasi-trees correspond
bijectively to spanning trees of the graph obtained by checkerboard coloring
$L$. This correspondence preserves the bigrading used for the spanning tree
model of Khovanov homology, whose Euler characteristic is the Jones polynomial
of $L$. Thus, Khovanov homology can be expressed in terms of ribbon graphs,
with generators given by ordered chord diagrams.
