Stability version of Dirac's theorem and its applications for generalized Turán problems

In 1952, Dirac proved that every $2$-connected $n$-vertex graph with the minimum degree $k+1$ contains a cycle of length at least $\min \{n,2(k+1)\}$. Here we obtain a stability version of this result by characterizing those graphs with minimum degree $k$ and circumference at most $2k+1$. We present applications of the above-stated result by obtaining generalized Turán numbers. In particular, for all $\mathrm{\ell }\ge 5$ we determine how many copies of a five-cycle as well as four-cycle are necessary to guarantee that the graph has circumference larger than $\mathrm{\ell }$. In addition, we give a new proof of Luo’s Theorem for cliques using our stability result.