On Dirac's Conjecture

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.