The famous Erdős–Gallai theorem on the Turán number of paths states that every graph with n vertices and m edges contains a path with at least (2m)/n edges. In this note, we first establish a simple but novel extension of the Erdős–Gallai theorem by proving that every graph G contains a path with at least $${{(s + 1){N_{s + 1}}(G)} \over {{N_s}(G)}} + s - 1$$ edges, where Nj(G) denotes the number of j-cliques in G for 1≤ j ≤ ω(G). We also construct a family of graphs which shows our extension improves the estimate given by the Erdős–Gallai theorem. Among applications, we show, for example, that the main results of [20], which are on the maximum possible number of s-cliques in an n-vertex graph without a path with ℓ vertices (and without cycles of length at least c), can be easily deduced from this extension. Indeed, to prove these results, Luo [20] generalized a classical theorem of Kopylov and established a tight upper bound on the number of s-cliques in an n-vertex 2-connected graph with circumference less than c. We prove a similar result for an n-vertex 2-connected graph with circumference less than c and large minimum degree. We conclude this paper with an application of our results to a problem from spectral extremal graph theory on consecutive lengths of cycles in graphs.

中文翻译：

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Erdős-Gallai 定理和罗定理的扩展

著名的关于 Turán 路径数的 Erdős-Gallai 定理指出，每个图n顶点和米边缘包含至少 (2米)/n边缘。在这篇笔记中，我们首先通过证明每个图G包含至少一条路径$${{(s + 1){N_{s + 1}}(G)} \over {{N_s}(G)}} + s - 1$$边缘，其中ñj(G) 表示的数量j- 派系G1≤ j ≤ ω(G). 我们还构建了一组图，显示我们的扩展改进了 Erdős-Gallai 定理给出的估计。在应用程序中，我们展示了，例如，[20] 的主要结果是关于最大可能数量的s- 派系n- 没有具有 ℓ 个顶点的路径的顶点图（并且至少没有长度循环C)，可以很容易地从这个扩展中推导出来。事实上，为了证明这些结果，罗 [20] 推广了 Kopylov 的一个经典定理，并建立了一个严格的上界s- 派系n- 周长小于的顶点 2 连通图C. 我们证明了一个类似的结果n- 周长小于的顶点 2 连通图C和大的最小度数。我们通过将我们的结果应用到谱极值图论中关于图中连续长度的循环的问题来结束本文。