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arXiv · 2008.03730

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

Shimon Kogan
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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 $β(G)$ 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 $β_d(G)$ 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 $β_0(G) = β(G)$.

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Shimon Kogan. 2021-01-07. A note on a Caro-Wei bound for the bipartite independence number in graphs. https://arxiv.org/abs/2008.03730

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