Overview
This study focuses on identifying long nontrivial cycles within Hamiltonian graphs. Specifically, it addresses graphs with $n$ vertices that contain a Hamiltonian cycle and have a minimum degree of at least 3. The research presents an improved bound for the length of an additional cycle present in such graphs.
Research Context
Prior work by Girão, Kittipassorn, and Narayanan (2019), published in Israel J. Math., established that an $n$-vertex Hamiltonian graph with a minimum degree of at least 3 contains a second cycle of length at least $n-O(n^{4/5})$. The current research builds upon this finding by seeking a tighter lower bound for the length of this nontrivial cycle.
Approach
The methodology employed for this research integrates two distinct approaches: a constructive method and a nonconstructive method. The constructive method specifically incorporates a result derived from poset theory. The combination of these methods was utilized to derive the improved bound for the cycle length. This combined approach informed the determination of the new lower bound.
Findings
The central finding of this research is an improved lower bound for the length of a nontrivial cycle in an $n$-vertex Hamiltonian graph with a minimum degree of at least 3. The study establishes that such a graph contains another cycle of length at least $n-O(n^{2/3})$. This new bound represents an improvement over the previously established bound of $n-O(n^{4/5})$. The researchers indicate that the obtained bound is the optimal achievable bound when considering the specific combination of constructive and nonconstructive methods employed in this study.
Why This Matters
This research refines our understanding of the structural properties of Hamiltonian graphs, particularly concerning the existence and length of multiple cycles. Establishing tighter bounds contributes to the theoretical framework of graph theory, providing more precise characterizations of these complex structures.