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A survey on the Chvátal-Erdös theorem

In 1857, the Irish mathematician Sir William Hamilton(1805-1865) invented a game of travelling around the edges of a graph from vertex to vertex. Hamilton described the game, in a letter to his friend Graves, as a mathematical game on the dodecahedron in which one person sticks five pins in any five consecutive vertices and the other is required to complete the path to form a spanning cycle. In other words, given a particular graph to begin with, the objective of the game was to find a path in the graph which passes through each vertex exactly once. Such a graph is called a Hamiltonian graph. For a long time there was no elegant characterization of Hamiltonian graphs, although several necessary and sufficient conditions were known. Interested readers may consult graph theory books by Bondy and Murty[3], Harary [10] and Diestel [7] for relevant materials. First, let us present a useful necessary condition: