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Extensions of Erdős-Gallai Theorem and Luo's Theorem with Applications

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 $\frac{2m}{n}$ 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 $\frac{(s+1)N_{s+1}(G)}{N_{s}(G)}+s-1$ edges, where $N_j(G)$ denotes the number of $j$-cliques in $G$ for $1\leq j\leqω(G)$. We also construct a family of graphs which shows our extension improves the estimate given by Erdős-Gallai Theorem. Among applications, we show, for example, that the main results of \cite{L17}, 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 \cite{L17} 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.