On some intriguing problems in Hamiltonian graph theory -- A survey

We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, $t$-tough graphs, and claw-free graphs

Subgraphs, Closures and Hamiltonicity

Closure theorems in hamiltonian graph theory are of the following type: Let G be a 2- connected graph and let u, v be two distinct nonadjacent vertices of G. If condition c(u,v) holds, then G is hamiltonian if and only if G + uv is hamiltonian. We discuss several results of this type in which u and v are vertices of a subgraph H of G on four vertices and c(u, v) is a condition on the neighborhoods of the vertices of H (in G). We also discuss corresponding sufficient conditions for hamiltonicity of G

The Quantum PCP Conjecture

The classical PCP theorem is arguably the most important achievement of classical complexity theory in the past quarter century. In recent years, researchers in quantum computational complexity have tried to identify approaches and develop tools that address the question: does a quantum version of the PCP theorem hold? The story of this study starts with classical complexity and takes unexpected turns providing fascinating vistas on the foundations of quantum mechanics, the global nature of entanglement and its topological properties, quantum error correction, information theory, and much more; it raises questions that touch upon some of the most fundamental issues at the heart of our understanding of quantum mechanics. At this point, the jury is still out as to whether or not such a theorem holds. This survey aims to provide a snapshot of the status in this ongoing story, tailored to a general theory-of-CS audience.Comment: 45 pages, 4 figures, an enhanced version of the SIGACT guest column from Volume 44 Issue 2, June 201

Boundary Hamiltonian theory for gapped topological phases on an open surface

In this paper we propose a Hamiltonian approach to gapped topological phases on an open surface with boundary. Our setting is an extension of the Levin-Wen model to a 2d graph on the open surface, whose boundary is part of the graph. We systematically construct a series of boundary Hamiltonians such that each of them, when combined with the usual Levin-Wen bulk Hamiltonian, gives rise to a gapped energy spectrum which is topologically protected; and the corresponding wave functions are robust under changes of the underlying graph that maintain the spatial topology of the system. We derive explicit ground-state wavefunctions of the system and show that the boundary types are classified by Morita-equivalent Frobenius algebras. We also construct boundary quasiparticle creation, measuring and hopping operators. These operators allow us to characterize the boundary quasiparticles by bimodules of Frobenius algebras. Our approach also offers a concrete set of tools for computations. We illustrate our approach by a few examples.Comment: 21 pages;references correcte