For positive integers $n>d\ge k$, let $\varphi (n,d,k)$ denote the least integer ϕ such that every n-vertex graph with at least ϕ vertices of degree at least d contains a path on $k+1$ vertices. Many years ago, Erdős, Faudree, Schelp and Simonovits proposed the study of the function $\varphi (n,d,k)$, and conjectured that for any positive integers $n>d\ge k$, it holds that $\varphi (n,d,k)\le \lfloor \frac{k-1}{2}\rfloor \lfloor \frac{n}{d+1}\rfloor +ϵ$, where $ϵ=1$ if k is odd and $ϵ=2$ otherwise. In this paper we determine the values of the function $\varphi (n,d,k)$ exactly. This confirms the above conjecture of Erdős et al. for all positive integers $k\ne 4$ and in a corrected form for the case $k=4$. Our proof utilizes, among others, a lemma of Erdős et al. [3], a theorem of Jackson [6], and a (slight) extension of a very recent theorem of Kostochka, Luo and Zirlin [7], where the latter two results concern maximum cycles in bipartite graphs. Moreover, we construct examples to provide answers to two closely related questions raised by Erdős et al.

对于正整数 $n>d\ge 克$， 让 $\phi (n,d,克)$表示最小整数ϕ使得每个n顶点图至少有ϕ个顶点的度数至少为d包含一条路径$克+1$顶点。许多年前，Erdős、Faudree、Schelp 和 Simonovits 提出了对函数的研究$\phi (n,d,克)$，并推测对于任何正整数 $n>d\ge 克$，它认为 $\phi (n,d,克)\le \lfloor \frac{克-1}{2}\rfloor \lfloor \frac{n}{d+1}\rfloor +\epsilon$， 在哪里 $\epsilon =1$如果k是奇数并且$\epsilon =2$除此以外。在本文中，我们确定函数的值$\phi (n,d,克)$确切地。这证实了 Erdős 等人的上述猜想。对于所有正整数$克\ne 4$ 并以案例的更正形式 $克=4$. 我们的证明利用了 Erdős 等人的引理等。[3]，Jackson 定理 [6]，以及 Kostochka、Luo 和 Zirlin [7] 最新定理的（轻微）扩展，其中后两个结果涉及二部图中的最大循环。此外，我们构建了示例来为 Erdős 等人提出的两个密切相关的问题提供答案。