A Note on Large Cycles in Graphs Around Conjectures of Bondy and Jung

TL;DR

This paper introduces new conditions and bounds related to large cycles in graphs, advancing understanding of Bondy's and Jung's conjectures by providing improved sufficient criteria and lower bounds for cycle lengths.

Contribution

It presents novel sufficient conditions for generalized cycles in k-connected graphs and establishes new lower bounds for the circumference under reverse hypotheses.

Findings

Proves Bondy's conjecture for certain variants under new conditions.

Provides improved lower bounds for the circumference in graphs.

Enhances understanding of cycle structures in k-connected graphs.

Abstract

Two new sufficient conditions for generalized cycles (including Hamilton and dominating cycles as special cases) in an arbitrary k-connected graph (k=1,2,...) are derived, which prove the truth of Bondy's (1980) famous conjecture for some variants significantly improving the result expected by the given hypothesis. Similarly, two new lower bounds for the circumference (the length of a longest cycle) are established for the reverse hypothesis proposed by Jung (2001).