On the average size of independent sets in triangle-free graphs

Abstract

Abstract. We prove an asymptotically tight lower bound on the average size of independent sets in a triangle-free graph on n vertices with maximum degree d, showing that an independent set drawn uniformly at random from such a graph has expected size at leastn. This gives an alternative proof of Shearer's upper bound on the Ramsey number R(3, k). We then prove that the total number of independent sets in a triangle-free graph with maximum degree d is at least expn . The constant 1/2 in the exponent is best possible. In both cases, tightness is exhibited by a random d-regular graph.Both results come from considering the hard-core model from statistical physics: a random independent set I drawn from a graph with probability proportional to λ |I| , for a fugacity parameter λ > 0. We prove a general lower bound on the occupancy fraction (normalized expected size of the random independent set) of the hard-core model on triangle-free graphs of maximum degree d. The bound is asymptotically tight in d for all λ = O d (1).We conclude by stating several conjectures on the relationship between the average and maximum size of an independent set in a triangle-free graph and give some consequences of these conjectures in Ramsey theory.

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

“…This method builds on previous work on independent sets and matchings and the Widom‐Rowlinson model , but here we generalize the previous approach in two ways: (1) we deal with q ‐spin models instead of 2‐spin models; (2) we deal with soft and hard constraints instead of just hard constraints. The family of linear programs in for matchings was an infinite family of LP's indexed by two parameters — the vertex degree d and a fugacity parameter — and the entire family could be solved analytically with a single proof via LP duality.…”

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

“…The general form of the method used here has been applied to problems of a similar nature; in [4,6,20] bounds on the number of independent sets or matchings in certain classes of graphs are obtained via bounds on an observable of the hard-core or monomer-dimer models, and a similar problem for the Widom-Rowlinson model was solved in [3]. One can also obtain tight bounds on the individual coefficients of the partition function in certain cases [7].…”

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