Bisections of graphs under degree constraints

In this paper, we investigate the problem of finding {\it bisections} (i.e., balanced bipartitions) in graphs. We prove the following two results for {\it all} graphs $G$: (1). $G$ has a bisection where each vertex $v$ has at least $(1/4 - o(1))d_G(v)$ neighbors in its own part; (2). $G$ also has a bisection where each vertex $v$ has at least $(1/4 - o(1))d_G(v)$ neighbors in the opposite part. These results are asymptotically optimal up to a factor of $1/2$, aligning with what is expected from random constructions, and provide the first systematic understanding of bisections in general graphs under degree constraints. As a consequence, we establish for the first time the existence of a function $f(k)$ such that for any $k\geq 1$, every graph with minimum degree at least $f(k)$ admits a bisection where every vertex has at least $k$ neighbors in its own part, as well as a bisection where every vertex has at least $k$ neighbors in the opposite part. Using a more general setting, we further show that for any $\varepsilon > 0$, there exist $c_\varepsilon, c'_\varepsilon > 0$ such that any graph $G$ with minimum degree at least $c_\varepsilon k$ (respectively, $c'_\varepsilon k$) admits a bisection satisfying: every vertex has at least $k$ neighbors in its own part (respectively, in the opposite part), and at least $(1 - \varepsilon)|V(G)|$ vertices have at least $k$ neighbors in the opposite part (respectively, in their own part). These results extend and strengthen classical graph partitioning theorems of Erd\H{o}s, Thomassen, and K\"{u}hn-Osthus, while additionally satisfying the bisection requirement.