A Lower Bound for the Circumference Involving Connectivity

TL;DR

This paper establishes a new lower bound for the circumference of a graph involving minimum degree, connectivity, and the length of the longest cycle in the graph minus a longest cycle, advancing understanding of cycle length bounds.

Contribution

It introduces a novel lower bound for the circumference that explicitly involves minimum degree, connectivity, and the length of the longest cycle in the graph's complement of a longest cycle.

Findings

New lower bound involving δ, κ, and c̄

Bound increases with δ, κ, and c̄

Advances cycle length estimation methods

Abstract

Let $G$ be a graph, $C$ a longest cycle in $G$ and $\overline{p}$, $\overline{c}$ the lengths of a longest path and a longest cycle in $G\backslash C$, respectively. Almost all lower bounds for the circumference base on a standard procedure: choose an initial cycle $C_0$ in $G$ and try to enlarge it via structures of $G\backslash C_0$ and connections between $C_0$ and $G\backslash C_0$ closely related to $\overline{p}$, $\overline{c}$ and connectivity $\kappa$. Actually, each lower bound obtained in result of this procedure, somehow or is related to $\kappa$, $\overline{p}$, $\overline{c}$ but in forms of various particular values of $\kappa$, $\overline{p}$, $\overline{c}$ and the major problem is to involve these invariants into such bounds as parameters. In this paper we present a lower bound for the circumference involving $\delta$, $\kappa$ and $\overline{c}$ and increasing with $\delta$, $\kappa$ and $\overline{c}$.