## Section: New Results

### Linear lambda terms as invariants of rooted trivalent maps

Participant : Noam Zeilberger.

Recent studies of the combinatorics of linear lambda calculus have uncovered some unexpected connections to the old and well-developed theory of graphs embedded on surfaces (also known as “maps”) [47], [87], [88].
In [19], we aimed to give a simple and conceptual account for one of these connections, namely the correspondence (originally described by Bodini, Gardy, and Jacquot [47]) between $\alpha $-equivalence classes of closed linear lambda terms and isomorphism classes of rooted trivalent maps on compact oriented surfaces without boundary.
One immediate application of this new account was a characterization of trivalent maps which are *bridgeless* (in the graph-theoretic sense of having no disconnecting edge) as linear lambda terms with no closed proper subterms.
In turn, this lead to a surprising but natural reformulation of the
Four Color Theorem as a statement about typing in lambda calculus.