Tutte Short Exact Sequences of Graphs

We associate two modules, the $G$-parking critical module and the toppling critical module, to an undirected connected graph $G$. We establish a Tutte-like short exact sequence relating the modules associated to $G$, an edge contraction $G/e$ and edge deletion $G \setminus e$ ($e$ is a non-bridge). As applications of these short exact sequences, we relate the vanishing of certain combinatorial invariants (the number of acyclic orientations on connected partition graphs satisfying a unique sink property) of $G/e$ to the equality of corresponding invariants of $G$ and $G \setminus e$. We also obtain a short proof of a theorem of Merino that the critical polynomial of a graph is an evaluation of its Tutte polynomial.Comment: 40 pages, 3 figure

Monomials, Binomials, and Riemann-Roch

The Riemann-Roch theorem on a graph G is related to Alexander duality in combinatorial commutive algebra. We study the lattice ideal given by chip firing on G and the initial ideal whose standard monomials are the G-parking functions. When G is a saturated graph, these ideals are generic and the Scarf complex is a minimal free resolution. Otherwise, syzygies are obtained by degeneration. We also develop a self-contained Riemann-Roch theory for artinian monomial ideals.Comment: 18 pages, 2 figures, Minor revision

Embeddings and immersions of tropical curves

We construct immersions of trivalent abstract tropical curves in the Euclidean plane and embeddings of all abstract tropical curves in higher dimensional Euclidean space. Since not all curves have an embedding in the plane, we define the tropical crossing number of an abstract tropical curve to be the minimum number of self-intersections, counted with multiplicity, over all its immersions in the plane. We show that the tropical crossing number is at most quadratic in the number of edges and this bound is sharp. For curves of genus up to two, we systematically compute the crossing number. Finally, we use our immersed tropical curves to construct totally faithful nodal algebraic curves via lifting results of Mikhalkin and Shustin.Comment: 23 pages, 14 figures, final submitted versio