On Longest Cycle $C$ of a graph $G$ via Structures of $G-C$

TL;DR

This paper establishes two sharp lower bounds for the length of the longest cycle in a graph based on the structure of the graph after removing that cycle, involving the longest path and cycle in the residual graph.

Contribution

It introduces new bounds relating the longest cycle length to the residual graph's structure and minimum degree, advancing cycle theory in graphs.

Findings

Derived bounds improve understanding of cycle lengths in graphs.

Bounds are sharp, indicating optimality.

Provides tools for analyzing cycle structures in complex graphs.

Abstract

Two sharp lower bounds for the length of a longest cycle $C$ of a graph $G$ are presented in terms of the lengths of a longest path and a longest cycle of $G-C$, denoted by $\overline{p}$ and $\overline{c}$, respectively, combined with minimum degree $\delta$: (1) $|C|\geq(\overline{p}+2)(\delta-\overline{p})$ and (2) $|C|\geq(\overline{c}+1)(\delta-\overline{c}+1)$.