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Dimers and cluster integrable systems

Alexander B. Goncharov, Richard Kenyon (2013)

Annales scientifiques de l'École Normale Supérieure

We show that the dimer model on a bipartite graph Γ on a torus gives rise to a quantum integrable system of special type, which we call acluster integrable system. The phase space of the classical system contains, as an open dense subset, the moduli space Ł Γ of line bundles with connections on the graph Γ . The sum of Hamiltonians is essentially the partition function of the dimer model. We say that two such graphs Γ 1 and Γ 2 areequivalentif the Newton polygons of the corresponding partition functions...

Distance in graphs

Roger C. Entringer, Douglas E. Jackson, D. A. Snyder (1976)

Czechoslovak Mathematical Journal

Dynamic WWW structures in 3 D .

Brandes, Ulrik, Kääb, Vanessa, Löh, Andres, Wagner, Dorothea, Willhalm, Thomas (2000)

Journal of Graph Algorithms and Applications

Edge-choosability and total-choosability of planar graphs with no adjacent 3-cycles

Daniel W. Cranston (2009)

Discussiones Mathematicae Graph Theory

Let G be a planar graph with no two 3-cycles sharing an edge. We show that if Δ(G) ≥ 9, then χ'ₗ(G) = Δ(G) and χ''ₗ(G) = Δ(G)+1. We also show that if Δ(G) ≥ 6, then χ'ₗ(G) ≤ Δ(G)+1 and if Δ(G) ≥ 7, then χ''ₗ(G) ≤ Δ(G)+2. All of these results extend to graphs in the projective plane and when Δ(G) ≥ 7 the results also extend to graphs in the torus and Klein bottle. This second edge-choosability result improves on work of Wang and Lih and of Zhang and Wu. All of our results use the discharging method...

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