2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/120636In this paper we prove a sufficient condition for the existence of a Hamilton cycle, which is applicable to a wide variety of graphs, including relatively sparse graphs. In contrast to previous criteria, ours is based on only two properties: one requiring expansion of ``small'' sets, the other ensuring the existence of an edge between any two disjoint ``large'' sets. We also discuss applications in positional games, random graphs and extremal graph theory.19 pagesCombinatorics05C45, 05C38, 05C80text