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

Settling Kahn's conjecture (2001), we prove the following upper bound on the number i(G) of independent sets in a graph G without isolated vertices:where du is the degree of vertex u in G. Equality occurs when G is a disjoint union of complete bipartite graphs. The inequality was previously proved for regular graphs by Kahn and Zhao.We also prove an analogous tight lower bound:where equality occurs for G a disjoint union of cliques. More generally, we prove bounds on the weighted versions of these quantities, i.e., the independent set polynomial, or equivalently the partition function of the hard-core model with a given fugacity on a graph.

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“…where in the last step we use that each u ∈ V 2 is contained in exactly f u edges of E 3 . Thus, to prove (9), it suffices to show…”

Section: Thus (8) Expands Asmentioning

confidence: 99%

“…Zhao's bipartite swapping trick [27,28] did not extend to q-colorings. Very recently, the d = 3 case was proved by Davies, Jenssen, Perkins, and Roberts [12] using the occupancy method (along with a computer-aided verification), and it was later extended to d = 4 [9].…”

Section: 3mentioning

confidence: 99%

The number of independent sets in an irregular graph

Sah

Sawhney

Stoner

et al. 2019

Journal of Combinatorial Theory, Series B

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“…Using entropy methods, they proved this conjecture under the additional assumption that G is bipartite. This conjecture was later resolved using a linear programming relaxation for 3-regular graphs by Davies et al [11] and for 4-regular graphs when k ≥ 5 by Davies [10].…”

Section: Discussionmentioning

confidence: 99%

A proof of Tomescu's graph coloring conjecture

Fox

Manners

2019

Journal of Combinatorial Theory, Series B

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In 1971, Tomescu conjectured that every connected graph G on n vertices with chromatic number k ≥ 4 has at most k!(k − 1) n−k proper k-colorings. Recently, Knox and Mohar proved Tomescu's conjecture for k = 4 and k = 5. In this paper, we complete the proof of Tomescu's conjecture for all k ≥ 4, and show that equality occurs if and only if G is a k-clique with trees attached to each vertex.

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“…Since G × K 2 is bipartite, it follows by Theorem 1.11 that every such H satisfies Conjecture 1.15 (see Section 3.4). An example of such H is given by the adjacency matrix , and for these H, the conjecture has been verified for 3-regular [16] and 4-regular [14] graphs G via the occupancy method with computer assistance.…”

Section: Graphical Brascamp-lieb Inequalitiesmentioning

confidence: 95%

“…14. A 2-spin model is biclique-maximizing if it is antiferromagnetic and cliquemaximizing if it is ferromagnetic.…”

mentioning

confidence: 99%

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

Cited by 5 publications

References 27 publications

The number of independent sets in an irregular graph

The number of independent sets in an irregular graph

A proof of Tomescu's graph coloring conjecture

A reverse Sidorenko inequality

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