Hamiltonian cycles in regular graphs


This chapter presents the theorem of Hamiltonian cycles in regular graphs. If in a graph of order n every vertex has degree at least 1/2n then the graph contains a Hamiltonian cycle. This theorem is the first in a long line of results concerning forcibly Hamiltonian degree sequences—that is, degree sequences all whose realizations are Hamiltonian. A question is discussed whether a two-connected (m – k)-regular graph G of order 2m is Hamiltonian if k(≥1) is sufficiently small. If instead of regularity we ask only that the minimal degree is m – 1 then the answer is negative. The order of k in the example above is best possible: the graph has to be Hamiltonian if k < c1m – c2 for some positive constants c1, c2. It seems very likely that the best value of c1 is 1/3. © 1978 North-Holland Publishing Company

Publication Title

Annals of Discrete Mathematics