A well-known result due to Caro (1979) and Wei (1981) states that every graph $G$ has an independent set of size at least $\sum_{v\in V(G)} \frac{1}{d(v) + 1}$, where $d(v)$ denotes the degree of vertex $v$. Alon, Kahn, and Seymour (1987) showed the following generalization: For every $k\geq 0$, every graph $G$ has a $k$-degenerate induced subgraph with at least $\sum_{v \in V(G)}\min\{1, \frac {k+1}{d(v)+1}\}$ vertices. In particular, for $k=1$, every graph $G$ with no isolated vertices has an induced forest with at least $\sum_{v\in V(G)} \frac{2}{d(v) + 1}$ vertices. Akbari, Amanihamedani, Mousavi, Nikpey, and Sheybani (2019) conjectured that, if $G$ has minimum degree at least $2$, then one can even find an induced linear forest of that order in $G$, that is, a forest where each component is a path. In this paper, we prove this conjecture and show a number of related results. In particular, if there is no restriction on the minimum degree of $G$, we show that there are infinitely many ``best possible'' functions $f$ such that $\sum_{v\in V(G)} f(d(v))$ is a lower bound on the maximum order of a linear forest in $G$, and we give a full characterization of all such functions $f$.

翻译：Caro (1979) 和 Wei (1981) 的一个著名结论指出：每个图 $G$ 都包含一个独立集，其大小至少为 $\sum_{v\in V(G)} \frac{1}{d(v) + 1}$，其中 $d(v)$ 表示顶点 $v$ 的度。Alon、Kahn 和 Seymour (1987) 给出了以下推广：对于每个 $k\geq 0$，每个图 $G$ 都包含一个 $k$-退化的诱导子图，其顶点数至少为 $\sum_{v \in V(G)}\min\{1, \frac {k+1}{d(v)+1}\}$。特别地，当 $k=1$ 时，每个无孤立顶点的图 $G$ 都包含一个诱导森林，其顶点数至少为 $\sum_{v\in V(G)} \frac{2}{d(v) + 1}$。Akbari、Amanihamedani、Mousavi、Nikpey 和 Sheybani (2019) 猜想：如果 $G$ 的最小度至少为 2，那么甚至可以在 $G$ 中找到一个具有该阶数的诱导线性森林，即每个连通分支均为路的森林。本文证明了这一猜想，并给出了一系列相关结果。特别地，若对 $G$ 的最小度不加限制，我们证明存在无穷多个“最优”函数 $f$，使得 $\sum_{v\in V(G)} f(d(v))$ 是 $G$ 中线性森林最大阶数的下界，并给出了所有此类函数 $f$ 的完整刻画。