Bipartite regular graphs and shortness parameters

By constructing sequences of non-Hamiltonian graphs it is proved that (1) for k ⩾ 4, the class of k-connected k-valent bipartite graphs has shortness exponent less than one and (2) the class of cyclically 4-edge-connected trivalent bipartite graphs has shortness coefficient less than one

Hamiltonicity in multitriangular graphs

The family of 5-valent polyhedral graphs whose faces are all triangles or 3s-gons, s ≥ 9, is shown to contain non-hamiltonian graphs and to have a shortness exponent smaller than one

Shaken dynamics: an easy way to parallel MCMC

We define a Markovian parallel dynamics for a class of spin systems on general interaction graphs. In this dynamics, beside the usual set of parameters $J_{xy}$, the strength of the interaction between the spins $\sigma_x$ and $\sigma_y$, and $\lambda_x$, the external field at site $x$, there is an inertial parameter $q$ measuring the tendency of the system to remain locally in the same state. This dynamics is reversible with an explicitly defined stationary measure. For suitable choices of parameter this invariant measure concentrates on the ground states of the Hamiltonian. This implies that this dynamics can be used to solve, heuristically, difficult problems in the context of combinatorial optimization. We also study the dynamics on $\mathbb{Z}^2$ with homogeneous interaction and external field and with arbitrary boundary conditions. We prove that for certain values of the parameters the stationary measure is close to the related Gibbs measure. Hence our dynamics may be a good tool to sample from Gibbs measure by means of a parallel algorithm. Moreover we show how the parameter allow to interpolate between spin systems defined on different regular lattices.Comment: 5 figure

Embedding of Complete Graphs in Broken Chimera Graphs

In order to solve real world combinatorial optimization problems with a D-Wave quantum annealer it is necessary to embed the problem at hand into the D-Wave hardware graph, namely Chimera or Pegasus. Most hard real world problems exhibit a strong connectivity. For the worst case scenario of a complete graph, there exists an efficient solution for the embedding into the ideal Chimera graph. However, since real machines almost always have broken qubits it is necessary to find an embedding into the broken hardware graph. We present a new approach to the problem of embedding complete graphs into broken Chimera graphs. This problem can be formulated as an optimization problem, more precisely as a matching problem with additional linear constraints. Although being NP-hard in general it is fixed parameter tractable in the number of inaccessible vertices in the Chimera graph. We tested our exact approach on various instances of broken hardware graphs, both related to real hardware as well as randomly generated. For fixed runtime, we were able to embed larger complete graphs compared to previous, heuristic approaches. As an extension, we developed a fast heuristic algorithm which enables us to solve even larger instances. We compared the performance of our heuristic and exact approaches.Comment: 26 pages, 9 figures, 2 table

Zero temperature solutions of the Edwards-Anderson model in random Husimi Lattices

We solve the Edwards-Anderson model (EA) in different Husimi lattices. We show that, at T=0, the structure of the solution space depends on the parity of the loop sizes. Husimi lattices with odd loop sizes have always a trivial paramagnetic solution stable under 1RSB perturbations while, in Husimi lattices with even loop sizes, this solution is absent. The range of stability under 1RSB perturbations of this and other RS solutions is computed analytically (when possible) or numerically. We compute the free-energy, the complexity and the ground state energy of different Husimi lattices at the level of the 1RSB approximation. We also show, when the fraction of ferromagnetic couplings increases, the existence, first, of a discontinuous transition from a paramagnetic to a spin glass phase and latter of a continuous transition from a spin glass to a ferromagnetic phase.Comment: 20 pages, 10 figures (v3: Corrected analysis of transitions. Appendix proof fixed

Connectivity and Cycles in Graphs

https://digitalcommons.memphis.edu/speccoll-faudreerj/1199/thumbnail.jp