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.
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[edit] Formal statement
P.J. Heawood conjectured in 1890 that for a given genus g > 0, the minimum number of colors necessary to color all graphs drawn on an orientable surface of that genus (or equivalently to color the regions of any partition of the surface into simply connected regions) is given by
where 
is the floor function.
Replacing the genus by the Euler characteristic, we obtain a formula that covers both the orientable and non-orientable cases,
This relation holds, as Ringel and Youngs showed, for all surfaces except for the Klein bottle. Philip Franklin (1930) proved that the Klein bottle requires at most 6 colors, rather than 7 as predicted by the formula. The Franklin graph can be drawn on the Klein bottle in a way that forms six mutually-adjacent regions, showing that this bound is tight.
The upper bound, proved in Heawood's original short paper, is straightforward: by manipulating the Euler characteristic, one can show that any graph embedded in the surface must have at least one vertex of degree less than the given bound. If one removes this vertex, and colors the rest of the graph, the small number of edges incident to the removed vertex ensures that it can be added back to the graph and colored without increasing the needed number of colors beyond the bound. In the other direction, the proof is more difficult, and involves showing that in each case (except the Klein bottle) a complete graph with a number of vertices equal to the given number of colors can be embedded on the surface.
[edit] Example
The torus has g = 1, so χ = 0. Therefore, as the formula states, any subdivision of the torus into regions can be colored using at most seven colors. The illustration shows a subdivision of the torus in which each of seven regions are adjacent to each other region; this subdivision shows that the bound of seven on the number of colors is tight for this case. The boundary of this subdivision forms an embedding of the Heawood graph onto the torus.
[edit] References
- Franklin, P. (1934). "A six color problem". J. Math. Phys. 13: 363–379.
- Heawood, P. J. (1890). "Map colour theorem". Quart. J. Math. 24: 332–338.
- Ringel, G.; Youngs, J. W. T. (1968). "Solution of the Heawood map-coloring problem". Proc. Nat. Acad. Sci. USA 60 (2): 438–445. doi:10.1073/pnas.60.2.438. PMC 225066. PMID 16591648. MR0228378. http://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=225066.
