2022/10/27 by Harish Kishnani, Kishnani, Harish, Rijubrata Kundu +3 · 1 citation
Engineering · Mathematics · Computer Science · #graph theory and CDMA systems #Limits and Structures in Graph Theory #Coding theory and cryptography
paper · pdf · doi:10.48550/arxiv.2210.15354
Given integers k,l≥ 2, where either l is odd or k is even, let n(k,l) denote the largest integer n such that each element of An is a product of k many l-cycles. In 2008, M. Herzog, G. Kaplan and A. Lev conjectured that \lfloor (2kl)/(3) \rfloor ≤ n(k,l)≤ \lfloor (2kl)/(3)\rfloor+1. It is known that the conjecture holds when k=2,3,4. Moreover, it is also true when 3| l. In this article, we determine the exact value of n(k,l) when 3\nmid l and k≥ 5. As an immediate consequence, we get that n(k,l)<\lfloor (2kl)/(3)\rfloor when k≥ 5, which shows that the above conjecture is not true in general. In fact, the difference between the exact value of n(k,l) and the conjectured value grows linearly in terms of k. Our results also generalize the case of k=2,3,4.