On Dirac's Conjecture

TL;DR

This paper establishes new sharp bounds relating the circumference, longest path, and a vine length in 2-connected graphs, generalizing Dirac's conjecture and providing improved inequalities.

Contribution

It introduces more general bounds involving the vine length, extending Dirac's conjecture with precise inequalities based on graph parameters.

Findings

Derived bounds depend on the parity of the vine length m.

Generalized Dirac's conjecture as a corollary.

Provided sharp inequalities relating c, l, m, and y.

Abstract

Let $G$ be a 2-connected graph, $l$ be the length of a longest path in $G$ and $c$ be the circumference - the length of a longest cycle in $G$. In 1952, Dirac proved that $c>\sqrt{2l}$ and conjectured that $c\ge 2\sqrt{l}$. In this paper we present more general sharp bounds in terms of $l$ and the length $m$ of a vine on a longest path in $G$ including Dirac's conjecture as a corollary: if $c=m+y+2$ (generally, $c\ge m+y+2$) for some integer $y\ge 0$, then $c\ge\sqrt{4l+(y+1)^2}$ if $m$ is odd; and $c\ge\sqrt{4l+(y+1)^2-1}$ if $m$ is even.