Hamiltonicity of random graphs Probabilistic combinatorics: Hamiltonicity is a focal point of probabilistic combinatorics, the cross-relationship between structural graph phenomena and threshold phenomena. In this paper, the authors examine the appearance of Hamilton cycles in classical random graph models, especially, ErdosRenyi random graph G(n,p) and G(n,m). We consider probabilistic scenarios in which Hamiltonicity is a high probability phenomenon, focusing on sharp thresholding as the edge probability p is changed. The paper combines the theoretical limits with algorithmic methods of finding Hamiltonian cycles, such as randomized and greedy heuristics. The performance analysis is performed in different p regimes and transition periods between disconnected to Hamiltonian graphs are observed. Findings show that Hamiltonicity appears around p = (logn)/n, which agrees with the classical findings. Findings give the insights into the structural changes and algorithm feasibility, which can be used in theoretical knowledge and practical use in network design and reliability analysis.