Extensions of the Erdős–Gallai theorem and Luo’s theorem

The famous Erd\H{o}s-Gallai Theorem states that every graph with $n$ vertices and $m$ edges contains a path of length at least $\frac{2m}{n}$. In this note, we first establish a simple but novel extension of Erd\H{o}s-Gallai Theorem by proving that every graph $G$ contains a path of length at least $\frac{(s+1)N_{s+1}(G)}{N_{s}(G)}+s-1$, where $N_j(G)$ denotes the number of $j$-cliques in $G$ for $1\leq j\leq\omega(G)$. We also construct a family of graphs which shows our extension improves the estimate given by Erd\H{o}s-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 $P_k$ (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.