Some Triangulated Surfaces without Balanced Splitting

Let G be the graph of a triangulated surface $\Sigma$ of genus $g\geq 2$. A cycle of G is splitting if it cuts $\Sigma$ into two components, neither of which is homeomorphic to a disk. A splitting cycle has type k if the corresponding components have genera k and g-k. It was conjectured that G contains a splitting cycle (Barnette '1982). We confirm this conjecture for an infinite family of triangulations by complete graphs but give counter-examples to a stronger conjecture (Mohar and Thomassen '2001) claiming that G should contain splitting cycles of every possible type.Comment: 15 pages, 7 figure

The critical Z-invariant Ising model via dimers: the periodic case

We study a large class of critical two-dimensional Ising models namely critical Z-invariant Ising models on periodic graphs, example of which are the classical square, triangular and honeycomb lattice at the critical temperature. Fisher introduced a correspondence between the Ising model and the dimer model on a decorated graph, thus setting dimer techniques as a powerful tool for understanding the Ising model. In this paper, we give a full description of the dimer model corresponding to the critical Z-invariant Ising model. We prove that the dimer characteristic polynomial is equal (up to a constant) to the critical Laplacian characteristic polynomial, and defines a Harnack curve of genus 0. We prove an explicit expression for the free energy, and for the Gibbs measure obtained as weak limit of Boltzmann measures.Comment: 35 pages, 8 figure

Long path and cycle decompositions of even hypercubes

We consider edge decompositions of the $n$-dimensional hypercube $Q_n$ into isomorphic copies of a given graph $H$. While a number of results are known about decomposing $Q_n$ into graphs from various classes, the simplest cases of paths and cycles of a given length are far from being understood. A conjecture of Erde asserts that if $n$ is even, $\ell < 2^n$ and $\ell$ divides the number of edges of $Q_n$, then the path of length $\ell$ decomposes $Q_n$. Tapadia et al.\ proved that any path of length $2^mn$, where $2^m<n$, satisfying these conditions decomposes $Q_n$. Here, we make progress toward resolving Erde's conjecture by showing that cycles of certain lengths up to $2^{n+1}/n$ decompose $Q_n$. As a consequence, we show that $Q_n$ can be decomposed into copies of any path of length at most $2^{n}/n$ dividing the number of edges of $Q_n$, thereby settling Erde's conjecture up to a linear factor

Knots with small rational genus

If K is a rationally null-homologous knot in a 3-manifold M, the rational genus of K is the infimum of -\chi(S)/2p over all embedded orientable surfaces S in the complement of K whose boundary wraps p times around K for some p (hereafter: S is a p-Seifert surface for K). Knots with very small rational genus can be constructed by "generic" Dehn filling, and are therefore extremely plentiful. In this paper we show that knots with rational genus less than 1/402 are all geometric -- i.e. they may be isotoped into a special form with respect to the geometric decomposition of M -- and give a complete classification. Our arguments are a mixture of hyperbolic geometry, combinatorics, and a careful study of the interaction of small p-Seifert surfaces with essential subsurfaces in M of non-negative Euler characteristic.Comment: 38 pages, 3 figures; version 3 corrects minor typos; keywords: knots, rational genu

Lifting and Separation Procedures for the Cut Polytope

The max-cut problem and the associated cut polytope on complete graphs have been extensively studied over the last 25 years. However, little research has been conducted for the cut polytope on arbitrary graphs. In this study we describe new separation and lifting procedures for the cut polytope on such graphs. These procedures exploit algorithmic and structural results known for the cut polytope on complete graphs to generate valid, and sometimes facet defining, inequalities for the cut polytope on arbitrary graphs in a cutting plane framework. We report computational results on a set of well-established benchmark problems