Stability in Bondy's theorem on paths and cycles

TL;DR

This paper investigates the stability of Bondy's theorem on paths and cycles, proving new bounds for 2-connected non-hamiltonian graphs and linking these results to algorithmic and spectral graph theory problems.

Contribution

It establishes a stability version of Bondy's theorem, providing new cycle length bounds and implications for polynomial-time algorithms and extremal graph characterizations.

Findings

Proves that certain 2-connected graphs contain long cycles unless they belong to specific families.

Shows the stability result can imply a polynomial-time decision algorithm for long cycle existence.

Identifies extremal graphs related to wheels on odd vertices.

Abstract

In this paper, we study the stability result of a well-known theorem of Bondy. We prove that for any 2-connected non-hamiltonian graph, if every vertex except for at most one vertex has degree at least $k$, then it contains a cycle of length at least $2k+2$ except for some special families of graphs. Our results imply several previous classical theorems including a deep and old result by Voss. We point out our result on stability in Bondy's theorem can directly imply a positive solution (in a slight stronger form) to the following problem: Is there a polynomial time algorithm to decide whether a 2-connected graph $G$ on $n$ vertices has a cycle of length at least $\min\{2\delta(G)+2,n\}$. This problem originally motivates the recent study on algorithmic aspects of Dirac's theorem by Fomin, Golovach, Sagunov and Simonov, although a stronger problem was solved by them by completely different methods. Our theorem can also help us to determine all extremal graphs for wheels on odd number of vertices. We also discuss the relationship between our results and some previous problems and theorems in spectral graph theory and generalized Tur\'{a}n problem.