2022 Combinatorics Workshop

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

1 Oct 2022, 11:45

20m

Contributed talk Session 3

Speaker

Nika Salia (IBS Extremal Combinatorics and Probability Group)

Description

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 Tur\'an numbers.
In particular, for all $\ell \ge 5$ we determine how many copies of a five-cycle as well as four-cycle are necessary to guarantee that the graph has a circumference larger than $\ell$.
In addition, we give new proof of Luo's Theorem for cliques using our stability result.