Completely independent spanning trees in some Cartesian product graphs

Let $T_{1}, T_{2}, \dots, T_{k}$ be spanning trees of a graph $G$. For any two vertices $u, v$ of $G$, if the paths from $u$ to $v$ in these $k$ trees are pairwise openly disjoint, then we say that $T_{1}, T_{2}, \dots, T_{k}$ are completely independent. Hasunuma showed that there are two completely independent spanning trees in any 4-connected maximal planar graph, and that given a graph $G$, the problem of deciding whether there exist two completely independent spanning trees in $G$ is NP-complete. In this paper, we consider the number of completely independent spanning trees in some Cartesian product graphs such as $W_{m}\Box P_{n}, \ W_{m}\Box C_{n}, \ K_{m, n}\Box P_{r}, \ K_{m, n}\Box C_{r}, \ K_{m, n, r}\Box P_{s}, \ K_{m, n, r}\Box C_{s}$

Dimers, Tilings and Trees

Generalizing results of Temperley, Brooks, Smith, Stone and Tutte and others we describe a natural equivalence between three planar objects: weighted bipartite planar graphs; planar Markov chains; and tilings with convex polygons. This equivalence provides a measure-preserving bijection between dimer coverings of a weighted bipartite planar graph and spanning trees on the corresponding Markov chain. The tilings correspond to harmonic functions on the Markov chain and to ``discrete analytic functions'' on the bipartite graph. The equivalence is extended to infinite periodic graphs, and we classify the resulting ``almost periodic'' tilings and harmonic functions.Comment: 23 pages, 5 figure

Searching for a Connection Between Matroid Theory and String Theory

We make a number of observations about matter-ghost string phase, which may eventually lead to a formal connection between matroid theory and string theory. In particular, in order to take advantage of the already established connection between matroid theory and Chern-Simons theory, we propose a generalization of string theory in terms of some kind of Kahler metric. We show that this generalization is closely related to the Kahler-Chern-Simons action due to Nair and Schiff. In addition, we discuss matroid/string connection via matroid bundles and a Schild type action, and we add new information about the relationship between matroid theory, D=11 supergravity and Chern-Simons formalism.Comment: 28 pages, LaTex, section 6 and references adde