Stability version of Dirac's theorem and its applications for generalized Tur\'an problems

TL;DR

This paper extends Dirac's theorem by providing a stability version that characterizes near-extremal graphs and applies this to determine generalized Turán numbers for cycles, offering new insights into graph circumference and clique structures.

Contribution

It introduces a stability version of Dirac's theorem and applies it to solve generalized Turán problems for cycles and cliques.

Findings

Characterization of graphs with minimum degree $k$ and circumference at most $2k+1$.

Determination of Turán numbers for cycles of length at least 5.

New proof of Luo's Theorem for cliques.

Abstract

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 $\ell \geq 5$ 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 $\ell$. In addition, we give a new proof of Luo's Theorem for cliques using our stability result.