On maximum induced forests of the balanced bipartite graphs

Abstract

Let $B$ be a balanced bipartite graph with two parts, $V_1$ and $V_2$, each containing $n$ vertices, resulting in a total of $2n$ vertices. Recently, Wang and Wu conjectured that if the minimum degree of $B$, denoted as $\delta \left(B\right)$, is greater than or equal to $\frac{n}{2}+1$, then the largest order of an induced forest in $B$ is equal to $n+1$. In this paper, we prove this conjecture and show that the condition on the minimum degree cannot be relaxed in general terms. Furthermore, we determine that if $\delta \left(B\right)\ge \frac{n}{2}+1$, then any subset $S$ of vertices in $B$ that induces a forest of size $n+1$ will satisfy the conditions $min\{|S\cap V_1|,|S\cap V_2|\}=1$ when $n$ is odd, and $min\{|S\cap V_1|,|S\cap V_2|\}\in \{1,2,\frac{n}{2}\}$ when $n$ is even. Additionally, we identify infinitely many balanced bipartite graphs that meet these conditions.