有向图的精确二分界
Sharp Bisection Bounds for Digraphs
浏览论文内容
中文总结 AI 辅助
该研究证明了最小半度至少为d的m条弧有向图的精确二分有向割下界，确定了最优加性修正项，还给出了通用有向图二分有向割下界并验证其紧性。
中文摘要 AI 辅助
固定整数$d\ge1$。我们证明，对所有足够大的$m$，每个具有$m$条弧且最小半度至少为$d$的有向图$D$都存在一个二分划分$V(D)=V_1\cup V_2$，满足$\bigl||V_1|-|V_2|\bigr|\le1$，使得$\min\{e(V_1,V_2),e(V_2,V_1)\} \ge \frac{d(m+d+1)}{2(2d+1)}$。其中，对于不相交的顶点集$A,B\subseteq V(D)$，$e(A,B)$表示从$A$指向$B$的弧的数量。对每个$d$，该界对无穷多个$m$值取等，表明加性项$d+1$是最优的。这回答了Liu、Ma和Zu的一个问题，消除了最小半度界中的渐近误差，将结果强化到二分情形，并确定了最优加性修正项。我们还证明，每个具有$m$条弧的$n$顶点有向图都存在一个二分划分，其中两个有向割的大小都至少为$(m-n+1)/4$，且这个通用界对外出星图是紧的。
英文摘要
Fix an integer $d\ge1$. We prove that, for all sufficiently large $m$, every digraph $D$ with $m$ arcs and minimum semidegree at least $d$ admits a bisection $V(D)=V_1\cup V_2$ with $\bigl||V_1|-|V_2|\bigr|\le1$ such that \[ \min\{e(V_1,V_2),e(V_2,V_1)\} \ge \frac{d(m+d+1)}{2(2d+1)}. \] Here, for disjoint vertex sets $A,B\subseteq V(D)$, $e(A,B)$ denotes the number of arcs directed from $A$ to $B$. For each $d$, equality holds for infinitely many values of $m$, showing that the additive term $d+1$ is best possible. This answers a question of Liu, Ma and Zu by removing the asymptotic error in the minimum-semidegree bound, strengthens the result to the bisection setting, and determines the optimal additive correction. We also prove that every $n$-vertex digraph with $m$ arcs admits a bisection in which both directed cuts have size at least $(m-n+1)/4$, and that this universal bound is sharp for out-stars.
