Cycles of Many Lengths in Hamiltonian Graphs

The chapter delves into the intricate study of cycles in graphs, with a particular focus on Hamiltonian graphs—graphs containing cycles that pass through all vertices. Starting with the fundamental theorem of Dirac, the text explores various conditions that guarantee the existence of a Hamilton cycle and their implications on pancyclicity, the property of containing cycles of all possible lengths. Bondy's influential meta-conjecture serves as a cornerstone, suggesting that any non-trivial condition ensuring Hamiltonicity also guarantees pancyclicity. The chapter highlights notable results, such as Bondy's proof that Ore's condition for Hamiltonicity implies pancyclicity, and the work of Bauer and Schmeichel on sufficient conditions for pancyclicity. Additionally, it discusses the conjecture by Jacobson and Lehel regarding the minimum size of the cycle spectrum in k-regular Hamiltonian graphs and the challenges in proving this conjecture. The text also presents the author's main theorem, which resolves the conjecture asymptotically, and proposes intermediate steps towards a linear bound on the number of cycle lengths. Throughout, the chapter offers a detailed exploration of the methods and ideas used to tackle these complex problems, making it a valuable resource for specialists in the field.