Let C(G) denote the set of lengths of cycles of a graph G of order n and let G¯ denote the complement of G. We show that if n≥6, then C(G)∪C(G¯) contains all odd ℓ with 3≤ℓ≤co and all even ℓ with 4≤ℓ≤ce, where co and ce denote the maximum odd and the maximum even integer in C(G)∪C(G¯), respectively. From this we deduce that the set C(G)∪C(G¯) contains at least 2n3−2 integers, which is sharp.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.