  We address the following question: When a randomly chosen regular bipartite
multi--graph is drawn in the plane in the ``standard way'', what is the
distribution of its maximum size planar matching (set of non--crossing disjoint
edges) and maximum size planar subgraph (set of non--crossing edges which may
share endpoints)? The problem is a generalization of the Longest Increasing
Sequence (LIS) problem (also called Ulam's problem). We present combinatorial
identities which relate the number of r-regular bipartite multi--graphs with
maximum planar matching (maximum planar subgraph) of at most d edges to a
signed sum of restricted lattice walks in Z^d, and to the number of pairs of
standard Young tableaux of the same shape and with a ``descend--type''
property. Our results are obtained via generalizations of two combinatorial
proofs through which Gessel's identity can be obtained (an identity that is
crucial in the derivation of a bivariate generating function associated to the
distribution of LISs, and key to the analytic attack on Ulam's problem).
  We also initiate the study of pattern avoidance in bipartite multigraphs and
derive a generalized Gessel identity for the number of bipartite 2-regular
multigraphs avoiding a specific (monotone) pattern.
