AI 中文总结
研究无孤立顶点的\(n\)顶点图，利用特定二分法，得出其有至少\(n/5\)个顶点且所有度数均为奇数的导出子图，改进了该类图的通用下界。
AI 中文摘要
加拉伊证明每个图都可划分为两个集合，每个集合诱导出一个所有度数均为偶数的子图。我们证明若一个图存在一种二分法，使得每个顶点在相对部分有奇数个邻居，那么它可划分为两个集合，每个集合诱导出一个所有度数均为奇数的子图。因此，每个无孤立顶点的\(n\)顶点图都有一个至少\(n/5\)个顶点的所有度数均为奇数的导出子图，大幅改进了先前已知的通用下界。
英文摘要
Gallai proved that every graph can be partitioned into two sets, each inducing a subgraph with all degrees even. We show that if a graph admits a bipartition in which every vertex has an odd number of neighbors in the opposite part, then it can be partitioned into two sets, each inducing a subgraph with all degrees odd. Consequently, every $n$-vertex graph without isolated vertices has an induced subgraph with all degrees odd on at least $n/5$ vertices, substantially improving the previously known universal lower bound.
