Caro–Wei-type lower bound for hard-core occupancy

Prove that for every finite graph G and fugacity λ>0, the expected size of a hard-core random independent set satisfies E|X| ≥ Σ_{v∈V(G)} λ/[1+(deg(v)+1)λ].

Background

The paper defines f_λ(d)=λ/[1+(d+1)λ] and notes that the previously established maximum-degree occupancy bound is sharp for disjoint unions of complete graphs. Conjecture A proposes a stronger degree-sensitive inequality, summing the corresponding local contributions over all vertices. Because f_λ is convex, the conjecture would also imply an average-degree version and strengthen the paper’s earlier occupancy and partition-function bounds, as well as the classical Caro–Wei theorem.

References

We propose the following conjecture. With $f_\lambda$ as above, let $X$ be a random independent set in a graph $G=(V,E)$, chosen according to the hard-core model at fugacity $\lambda$. Then

— The hard-core model in graph theory  (2501.03379 - Davies et al., 6 Jan 2025) in Conjecture A, Section “Open problems”

It is less clear whether there is a canonical independence polynomial bound corresponding to Shearer's independence number bound for triangle-free graphs of given average degree, or, more in line with the Caro--Wei theorem, degree-sequence versions thereof.

— Random independent sets and local sparsity  (2609.04654 - Davies, 4 Sep 2026) in Section 1, Introduction, paragraph discussing lower bounds on the independence polynomial