Abstract

Given k≥3 and 1≤ℓ<k, an (ℓ,k)-cycle is one in which consecutive edges, each of size k, overlap in exactly ℓ vertices. We study the smallest number of edges in k-uniform n-vertex hypergraphs which do not contain hamiltonian (ℓ,k)-cycles, but once a new edge is added, such a cycle is promptly created. It has been conjectured that this number is of order nℓ and confirmed for ℓ∈{1,k/2,k−1}, as well as for the upper range 0.8k≤ℓ≤k−1. Here we extend the validity of this conjecture to the lower–middle range (k−1)/3≤ℓ<(k−1)/2.

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.