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

Abstract

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 \[ β(G) \geq \left \lfloor \frac{1}{2} \cdot \sum_{v \in V(G)} \frac{1}{d(v)+1} \right \rfloor. \] 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 \[ β_d(G) \geq \left \lfloor \frac{1}{2} \cdot \sum_{v \in V(G)} \min\left(1,\frac{d+1}{d(v)+1}\right) \right \rfloor. \] Notice that ${\beta }_0(G)=\beta (G)$.