Heawood Conjecture

In graph theory, the Heawood conjecture or Ringel–Youngs theorem gives a lower bound for the number of colors that are necessary for graph coloring on a surface of a given genus. It was proven in 1968 by Gerhard Ringel and John W. T. (Ted) Youngs. One case, the non-orientable Klein bottle, proved an exception to the general formula. An entirely different approach was needed for the much older problem of finding the number of colors needed for the plane, or equivalently, the sphere, solved in 1976 as the four color theorem by Haken and Appel. On the sphere the lower bound is easy, whereas for higher genera the upper bound is easy and was proved in Heawood's original short paper that contained the conjecture. In other words, Ringel, Youngs and others had to construct extreme examples for every genus g = 1,2,3,... If g = 12s + k, the genera fall into 12 Cases according as k = 0,1,2,3,4,5,6,7,8,9,10,11. To simplify the discussion, let's say that Case k has been established if only a finite number of g's of the form 12s + k are in doubt. Then the years in which the twelve Cases were settled and by whom are the following:

• 1954 Ringel Case 5
• 1961 Ringel Cases 3,7,10
• 1963 Terry, Welch, Youngs Cases 0,4
• 1964 Gustin, Youngs Case 1
• 1965 Gustin Case 9
• 1966 Youngs Case 6
• 1967 Ringel, Youngs Cases 2,8,11
• The last seven sporadic exceptions were settled as follows:
• 1967 Mayer 18, 20, 23
• 1968 Ringel, Youngs 30,35,47,59, and the conjecture was proved.
• 1970 Prof. John W. T. (Ted) Youngs persuaded his coauthor Gerhard Ringel to move to U.C. Santa Cruz.