Andrea Jiménez

Computational hardness of enumerating groundstates of the antiferromagnetic Ising model in triangulations

Satisfying spin-assignments of triangulations of a surface are states of minimum energy of the antiferromagnetic Ising model on triangulations which correspond (via geometric duality) to perfect matchings in cubic bridgeless graphs. In this work we show that it is NP -complete to decide whether or not a triangulation of a surface admits a satisfying spin-assignment, and that it is # P -complete to determine the number of such assignments. Our results imply that the determination of even the entropy of the Ising model on triangulations at the thermodynamical limit is already # P -hard.The aforementioned claims are derived via elaborate (and atypical) reductions that map a Boolean formula in conjunctive normal form into triangulations of orientable closed surfaces. Moreover, the novel reduction technique enables us to prove that even very constrained versions of # MaxCut are already # P -hard.

Counting perfect matchings in the geometric dual

Abstract Lovasz and Plummer conjectured, in the mid 1970ʼs, that every bridgeless cubic graph has exponentially many perfect matchings. In this work we show that every cubic planar graph G whose geometric dual graph is a stack triangulation (planar 3-tree) has at least 3 ϕ | V ( G ) | / 72 distinct perfect matchings, where ϕ is the golden ratio. Our work builds on a novel approach relating Lovasz and Plummerʼs claim and the number of so called groundstates of the widely studied Ising model.

On path decompositions of 2k-regular graphs☆

Tibor Gallai conjectured that the edge set of every connected graph

Directed cycle double covers: hexagon graphs

G

Maximum number of sum-free colorings in finite abelian groups

on

Cubic Bridgeless Graphs and Braces

n

Small degeneracy of antiferromagnetic triangulations

vertices can be partitioned into

Counting perfect matchings of cubic graphs in the geometric dual

\lceil n/2\rceil

On path-cycle decompositions of triangle-free graphs

paths. Let

On path decompositions of 2k-regular graphs

\mathcal{G}_{k}