A Common Generalization of Dirac's two Theorems

Abstract

Let $G$ be a 2-connected graph of order $n$ and let $c$ be the circumference - the order of a longest cycle in $G$. In this paper we present a sharp lower bound for the circumference based on minimum degree $\delta$ and $p$ - the order of a longest path in $G$. This is a common generalization of two earlier classical results for 2-connected graphs due to Dirac: (i) $c\ge \min\{n,2\delta\}$; and (ii) $c\ge\sqrt{2p}$. Moreover, the result is stronger than (ii).