A note on a Caro-Wei bound for the bipartite independence number in graphs

A bi-hole of size $t$ in a bipartite graph $G$ is a copy of $K_{t,t}$ in the bipartite complement of $G$. Given an $n \times n$ bipartite graph $G$, let $\beta(G)$ be the largest $k$ for which $G$ has a bi-hole of size $k$. We prove that $$\beta(G) \geq \left \lfloor \frac{1}{2} \cdot \sum_{v \in V(G)} \frac{1}{d(v)+1} \right \rfloor.$$ Furthermore, we prove the following generalization of the result above. Given an $n \times n$ bipartite graph $G$, Let $\beta_d(G)$ be the largest $k$ for which $G$ has a $k \times k$ $d$-degenerate subgraph. We prove that $$\beta_d(G) \geq \left \lfloor \frac{1}{2} \cdot \sum_{v \in V(G)} \min\left(1,\frac{d+1}{d(v)+1}\right) \right \rfloor.$$ Notice that $\beta_0(G) = \beta(G)$.