A cycle structure theorem for Hamiltonian graphs
An n-vertex graph is called pancyclic if it contains a cycle of length \(\ell\) for every \(\ell\) such that \(3\leq \ell \leq n.\) We establish the following result: Let C be a hamiltonian cycle in an n- vertex graph G. Suppose C contains consecutive vertices x,y such that \(d(x)+d(y)\geq n\). Then G is either (i) pancyclic, (ii) bipartite, or (iii) missing only an (n-1)-cycle. Moreover, (iii) can only occur if G has a very explicit structure ''near'' x and y. This result can be used to show that three well-known hamiltonian degree conditions (due to Chvátal, Fan, and Bondy) actually imply that a graph is essentially pancyclic.
scientific article; zbMATH DE number 3981217
scientific article