Speaker：Bo Ning (Nankai University)

Time：2024-12-23, 14:30

Location：Conference Room C610 at Administration Building at Haiyun Campus

Abstract:

In this talk, we discuss about the stability result of a well-known theorem of Bondy. We shall show 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. The last but not least, we shall discuss the relationship between our results and some previous problems and theorems in spectral graph theory and generalized Turán problem. In particular, we shall present some work proved by other authors which uses our results as some new tool.