Origami: The Ways of 3-Edges Coloring for a Dodecahedron - Math Problem Example

For a dodecahedron, at least 3 different colors are required, as a dodecahedron cannot be properly colored in less than 3 colors. It is advisable to draw the planar graph of a dodecahedron when planning the 3-edge coloring. "It is always quite puzzling to try to make use of only 3 colors of paper with no two units of the same color touching. Each unit corresponds to an edge of the planar graph, so this is equivalent to a proper 3-edge-coloring of the polyhedron." (T.Hull, 2006) During the nineteenth century, Sir William Rowan Hamilton who was a mathematician from Ireland invented a puzzle known as 'Around the World.

' The concept behind the puzzle was to label the vertices of a regular dodecahedron according to the names of various cities of the world. Hamilton's puzzle can be solved by beginning from any given city (i.e. any vertex) and traveling around the world from one city (vertex) to another. This entails that one moves along the edges of the dodecahedron in such a manner that each other city is touched only once before going back to the original vertex or starting point. This solution to Hamilton's puzzle is known as a Hamilton cycle/Hamilton circuit.

Thus, a Hamilton circuit can be said to be a path in the dodecahedron which starts at a vertex, touches every other vertex, in the dodecahedron, and then returns to the original starting point without touching any single vertex two times.