Tight bounds on the coefficients of partition functions via stability

Abstract

Abstract. We show how to use the recently-developed occupancy method and stability results that follow easily from the method to obtain extremal bounds on the individual coefficients of the partition functions over various classes of bounded degree graphs.As applications, we prove new bounds on the number of independent sets and matchings of a given size in regular graphs. For large enough graphs and almost all sizes, the bounds are tight and confirm the Upper Matching Conjecture of Friedland, Krop, and Markström, and a conjecture of Kahn on independent sets for a wide range of parameters. Additionally we prove tight bounds on the number of q-colorings of cubic graphs with a given number of monochromatic edges, and tight bounds on the number of independent sets of a given size in cubic graphs of girth at least 5.

Tight bounds on the coefficients of partition functions via stability

“…For the hard-core model a similar stability result for λ = 1 was given in [9], and with some work it should be possible to deduce stability in Bregman's theorem from Radhakrishnan's proof [14]. The main benefit of obtaining stability via the occupancy fraction is generality, but in applications [8] we make crucial use of the monotonicity of s(d, λ) in λ. The methods of [6] have been successfully applied to analogous problems on independent sets and colourings [4,7,13]; in each of these cases finding a unique connected extremal graph for the problem of maximising a derivative of the free energy of a probabilistic model.…”

“…Theorem 1 (Davies, Jenssen, Perkins, Roberts [8]). Let G be an n-vertex, d-regular graph which contains no copy of K d,d .…”

“…The methods of [6] have been successfully applied to analogous problems on independent sets and colourings [4,7,13]; in each of these cases finding a unique connected extremal graph for the problem of maximising a derivative of the free energy of a probabilistic model. Our general method for proving stability applies in each case, yielding a result of the same form, see [8].…”

“…In particular, for large enough graphs and almost all sizes, we give tight upper bounds on the number of independent sets and matchings of fixed size in regular graphs, confirming conjectures of Kahn [12] and Friedland, Krop, and Markström [11] for a wide range of parameters. We give details in a forthcoming paper [8], here choosing to focus on the example of matchings in regular graphs.…”

Hypergraph Matchings and Designs

“…Upper Matching Conjecture of Friedland, Krop and Markström[31] by Davies, Jenssen, Perkins and Roberts[19].…”