Extremal Regular Graphs: Independent Sets and Graph Homomorphisms

Zhao, Yufei

doi:10.4169/amer.math.monthly.124.9.827

Cited by 20 publications

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References 37 publications

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“…There is a strong link between the Potts model and extremal combinatorics, as in the limit β → +∞, the model converges to the uniform distribution on proper q-colourings of a graph, and Z q G (β) tends to c q (G), the number of proper q-colourings of G. Extremal questions on the number of q-colourings have received a lot of attention; see e.g. [14,15] for examples in dense graphs and [25] for a survey (which also covers related problems) on regular graphs. A conjecture of Galvin and Tetali [10] states that over all d-regular graphs G and q ≥ 3, we have…”

Section: Introductionmentioning

confidence: 99%

Counting Proper Colourings in 4-Regular Graphs via the Potts Model

Davies

2018

Electron. J. Combin.

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We give tight upper and lower bounds on the internal energy per particle in the antiferromagnetic q-state Potts model on 4-regular graphs, for q ≥ 5. This proves the first case of a conjecture of the author, Perkins, Jenssen, and Roberts, and implies tight bounds on the antiferromagnetic Potts partition function.The zero-temperature limit gives upper and lower bounds on the number of proper q-colourings of 4-regular graphs, which almost proves the case d = 4 of a conjecture of Galvin and Tetali. For any q ≥ 5 we prove that the number of proper q-colourings of a 4-regular graph is maximised by a union of K4,4's.

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Section: Introductionmentioning

confidence: 99%

Counting Proper Colourings in 4-Regular Graphs via the Potts Model

Davies

2018

Electron. J. Combin.

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Extremes of the internal energy of the Potts model on cubic graphs

Davies

Jenssen

Perkins

et al. 2018

Random Struct Algorithms

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We prove tight upper and lower bounds on the internal energy per particle (expected number of monochromatic edges per vertex) in the anti‐ferromagnetic Potts model on cubic graphs at every temperature and for all q≥2. This immediately implies corresponding tight bounds on the anti‐ferromagnetic Potts partition function. Taking the zero‐temperature limit gives new results in extremal combinatorics: the number of q‐colorings of a 3‐regular graph, for any q≥2, is maximized by a union of K3,3's. This proves the d = 3 case of a conjecture of Galvin and Tetali.

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A reverse Sidorenko inequality

et al. 2020

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Let H be a graph allowing loops as well as vertex-and edge-weights. We prove that, for every triangle-free graph G without isolated vertices, the weighted number of graph homomorphisms hom(G, H) satisfies the inequalitywhere du denotes the degree of vertex u in G. In particular, one has hom(G, H) 1/|E(G)| ≤ hom (K d,d , H) 1/d 2 for every d-regular triangle-free G. The triangle-free hypothesis on G is best possible. More generally, we prove a graphical Brascamp-Lieb type inequality, where every edge of G is assigned some two-variable function. These inequalities imply tight upper bounds on the partition function of various statistical models such as the Ising and Potts models, which includes independent sets and graph colorings.For graph colorings, corresponding to H = Kq (also valid if some of the vertices of Kq are looped), we show that the triangle-free hypothesis on G may be dropped. A corollary is that among d-regular graphs, G = K d,d maximizes the quantity cq(G) 1/|V (G)| for every q and d, where cq(G) counts proper q-colorings of G.Finally, we show that if the edge-weight matrix of H is positive semidefinite, thenThis implies that among d-regular graphs, G = K d+1 maximizes hom(G, H) 1/|V (G)| . For 2-spin Ising models, our results give a complete characterization of extremal graphs: complete bipartite graphs maximize the partition function of 2-spin antiferromagnetic models and cliques maximize the partition function of ferromagnetic models. These results settle a number of conjectures by Galvin-Tetali, Galvin, and Cohen-Csikvári-Perkins-Tetali, and provide an alternate proof to a conjecture by Kahn. arXiv:1809.09462v2 [math.CO] , where denotes a disjoint union. Question 1.1 was initially raised by Granville in 1988 in connection with the Cameron-Erdős conjecture on the number of sum-free sets. Alon [1] and Kahn [27] conjectured that G = K d,d is the exact maximizer. Alon [1] proved an asymptotic version as d → ∞, Kahn [27] proved the exact version under the additional hypothesis that G is bipartite, and Zhao [38] later removed this bipartite assumption. The results of Kahn [27] and Zhao [38] together answer Question 1.1: the maximizer is K d,d (unique up to taking disjoint unions of copies of itself). Galvin and Tetali [22] initiated the study of Questions 1.2 and 1.3 and extended Kahn's entropy method [27] to prove that, under the additional hypothesis that G is bipartite, G = K d,d is also the maximizer for hom(G, H) 1/|V (G)| . See Lubetzky and Zhao [32, Section 6] for a different proof using Hölder/Brascamp-Lieb type inequalities. Can the bipartite hypothesis on G also be dropped in this case? Not for all H: e.g., for H = , G = K d+1 is the maximizer instead of K d,d . Extending the technique for independent sets, Zhao [39] showed that the bipartite hypothesis can be dropped for certain classes of H, but the techniques failed for H = K q , corresponding to colorings (Question 1.2). It remained a tantalizing conjecture to remove the bipartite hypothesis for colorings.Recently, Davies, Jenssen, Perki...

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Extremal Regular Graphs: Independent Sets and Graph Homomorphisms

Abstract: Abstract. This survey concerns regular graphs that are extremal with respect to the number of independent sets, and more generally, graph homomorphisms. More precisely, in the family of of d-regular graphs, which graph G maximizes/minimizes the quantity i(G)

Cited by 20 publications

References 37 publications

Counting Proper Colourings in 4-Regular Graphs via the Potts Model

Counting Proper Colourings in 4-Regular Graphs via the Potts Model

Extremes of the internal energy of the Potts model on cubic graphs

A reverse Sidorenko inequality

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