On a property of $2$-connected graphs and Dirac's Theorem

TL;DR

This paper refines a property of 2-connected graphs from Dirac's 1952 work and uses it to provide a shorter proof that such graphs with minimum degree k contain a cycle of length at least min{n, 2k}.

Contribution

It introduces a refined property of 2-connected graphs and applies it to simplify Dirac's classical proof regarding cycle lengths.

Findings

Refined a classical property of 2-connected graphs.

Provided a shorter proof of Dirac's theorem on cycle lengths.

Confirmed the minimum cycle length in 2-connected graphs with given degree constraints.

Abstract

We refine a property of $2$-connected graphs described in the classical paper of Dirac from 1952 and use the refined property to somewhat shorten Dirac's proof of the fact that each $2$-connected $n$-vertex graph with minimum degree at least $k$ has a cycle of length at least $\min\{n,2k\}$.