Projective plane embeddings of polyhedral pinched maps

We give various conditions on pinched-torus polyhedral maps which are necessary for their graphs to be embeddable in the projective plane. Our other main result is that even if the graph of a polyhedral map in the pinched torus is embeddable in a projective plane, the map induced by the embedding cannot be polyhedral, but must have all faces bounded by cycles. Finally, we give a class of examples of graphs which have polyhedral embeddings on the pinched torus and also on orientable surfaces of arbitrary high genus

On Zero-One and Convergence Laws for Graphs Embeddable on a Fixed Surface

We show that for no surface except for the plane does monadic second-order logic (MSO) have a zero-one-law - and not even a convergence law - on the class of (connected) graphs embeddable on the surface. In addition we show that every rational in [0,1] is the limiting probability of some MSO formula. This strongly refutes a conjecture by Heinig et al. (2014) who proved a convergence law for planar graphs, and a zero-one law for connected planar graphs, and also identified the so-called gaps of [0,1]: the subintervals that are not limiting probabilities of any MSO formula. The proof relies on a combination of methods from structural graph theory, especially large face-width embeddings of graphs on surfaces, analytic combinatorics, and finite model theory, and several parts of the proof may be of independent interest. In particular, we identify precisely the properties that make the zero-one law work on planar graphs but fail for every other surface

A Turaev surface approach to Khovanov homology

We introduce Khovanov homology for ribbon graphs and show that the Khovanov homology of a certain ribbon graph embedded on the Turaev surface of a link is isomorphic to the Khovanov homology of the link (after a grading shift). We also present a spanning quasi-tree model for the Khovanov homology of a ribbon graph.Comment: 30 pages, 18 figures, added sections on virtual links and Reidemeister move

Independent paths and K5-subdivisions

A well known theorem of Kuratowski states that a graph is planar iff it contains no subdivision of K5 or K3,3. Seymour conjectured in 1977 that every 5-connected nonplanar graph contains a subdivision of K5. In this paper, we prove several results about independent paths (no vertex of a path is internal to another), which are then used to prove Seymour’s conjecture for two classes of graphs. These results will be used in a subsequent paper to prove Seymour’s conjecture for graphs containing K − 4, which is a step in a program to approach Seymour’s conjecture