Stability in Bondy's theorem on paths and cycles
Bo Ning, Long-tu Yuan
TL;DR
Proves a stability version of Bondy's theorem on long cycles for 2-connected non-Hamiltonian graphs: if every vertex except at most one has degree at least $k$ and $c(G)\le 2k+1$, then $G$ is a subgraph of one of the explicit extremal families $H(2k+1,2k+1,k)$, $H(n,2k+2,k)$, $F(s,t,k)$, $F_1(t,k)$, $K_2+M_t$, $K_2+(S_s\cup M_t)$, or $K_3+M_t$. The proof adapts Erdős–Gallai and Bondy–Jackson frameworks and employs a vine-based longest-path analysis to establish the structure. This stability result unifies and extends classical theorems (including Voss) and yields algorithmic consequences for detecting long cycles in near-regular, 2-connected graphs, as well as corollaries for odd-wheel extremal graphs and links to spectral and generalized Turán problems.
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 $\min\{2δ(G)+2,n\}$. 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 Turán problem.