New bounds for the number of lightest cycles in undirected graphs

Abstract

Consider an undirected graph $G=(V,E)$ with positive integer edge weights. Subramanian [11] established an upper bound of $|V{|}^4/6$ on the number of minimum weight cycles. We present a new algorithm to enumerate all minimum weight cycles with a complexity of $O\left(\right|V{|}^3\left(\right|E|+|V|\log |V\left|\right))$. Using this algorithm, we derive the following upper bounds for the number of minimum weight cycles: if the minimum weight is even, the bound is $|V{|}^4/4$, and if it is odd, the bound is $|V{|}^3/2$. Notably, we improve Subramanian's bound by an order of magnitude when the minimum weight of a cycle is odd. Additionally, we demonstrate that these bounds are asymptotically tight.