Hard-core Caro–Wei conjecture

For $\lambda>0$, define

$$f_\lambda(d)=\frac{\lambda}{1+(d+1)\lambda}.$$

Let $G=(V,E)$ be a graph, and let $X$ be a random independent set chosen from the hard-core model at fugacity $\lambda$.

Hard-core Caro–Wei conjecture. The expected size of $X$ satisfies

$$\mathbb{E}|X|\ge\sum_{v\in V}f_\lambda(\deg(v)).$$

This would strengthen the average-degree occupancy bound, remain sharp for disjoint unions of complete graphs, and refine the Caro–Wei theorem and a related lower bound on the hard-core partition function.