Advanced Lower Bounds for the Circumference

Abstract

The classic lower bounds δ + 1 (Dirac), 2δ (Dirac) and 3δ − 3 (Voss and Zuluaga) for the circumference (the order of a longest cycle C in a graph G) are based on the minimum degree δ and some G\C structures, combined with some additional connectivity conditions. A natural problem arises to find an analogous bound in a general form (i + 1)(δ − i + 1) with i = 0, 1, . . . , δ, including the bounds δ + 1, 2δ and 3δ − 3 as special cases. In this paper we present two tight lower bounds for the circumference just of the form (i + 1)(δ − i + 1) for each ${i\in\{0,\ldots,\delta\}}$ based entirely on appropriate G\C structures, actually escaping any other additional conditions: in each graph G, (i) ${|C|\geq(\overline{p}+1)(\delta-\overline{p}+1)}$ and (ii) ${|C|\geq(\overline{c}+1)(\delta-\overline{c}+1),}$ where ${\overline{p}}$ and ${\overline{c}}$ denote the orders of a longest path and a longest cycle in G\C, respectively.