# Heawood conjecture

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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. For surfaces of genus 0, 1, 2, 3, 4, 5, 6, 7, ..., the required number of colors is 4, 7, 8, 9, 10, 11, 12, 12, .... , the chromatic number or Heawood number.

In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices which are connected by edges. A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges, then called arrows, link two vertices asymmetrically; see Graph for more detailed definitions and for other variations in the types of graph that are commonly considered. Graphs are one of the prime objects of study in discrete mathematics.

In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color.

In topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space.

## Contents

The conjecture was formulated in 1890 by Percy John Heawood and proven in 1968 by Gerhard Ringel and 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 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, suppose 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:

Percy John Heawood was a British mathematician educated at Queen Elizabeth's School, Ipswich, and Exeter College, Oxford. He spent his career at Durham University, where he was appointed Lecturer in 1885. He was, successively, Censor of St Cuthbert's Society between 1897 and 1901 succeeding Frank Byron Jevons in the role, Senior Proctor of the university from 1901, Professor in 1910 and Vice-Chancellor between 1926 and 1928. He was awarded an OBE, as Honorary Secretary of the Preservation Fund, for his part in raising £120,000 to prevent Durham Castle from collapsing into the River Wear.

Gerhard Ringel was a German mathematician who earned his Ph.D. from the University of Bonn in 1951. He was one of the pioneers in graph theory and contributed significantly to the proof of the Heawood conjecture, a mathematical problem closely linked with the Four Color Theorem.

John William Theodore Youngs was an American mathematician.

• 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: cases 18, 20, 23
• 1968, Ringel, Youngs: cases 30, 35, 47, 59, and the conjecture was proved.

## Formal statement

Percy John 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

In mathematics, a conjecture is a conclusion or proposition based on incomplete information, for which no proof has been found. Conjectures such as the Riemann hypothesis or Fermat's Last Theorem have shaped much of mathematical history as new areas of mathematics are developed in order to prove them.

${\displaystyle \gamma (g)=\left\lfloor {\frac {7+{\sqrt {1+48g}}}{2}}\right\rfloor ,}$

where ${\displaystyle \left\lfloor x\right\rfloor }$ is the floor function.

Replacing the genus by the Euler characteristic, we obtain a formula that covers both the orientable and non-orientable cases,

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by .

${\displaystyle \gamma (\chi )=\left\lfloor {\frac {7+{\sqrt {49-24\chi }}}{2}}\right\rfloor .}$

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.

In topology, a branch of mathematics, the Klein bottle is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a surface with boundary, a Klein bottle has no boundary.

Philip Franklin was an American mathematician and professor whose work was primarily focused in analysis.

In the mathematical field of graph theory, the Franklin graph a 3-regular graph with 12 vertices and 18 edges.

The upper bound, proved in Heawood's original short paper, is based on a greedy coloring algorithm. By manipulating the Euler characteristic, one can show that every graph embedded in the given 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.

## 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.

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## References

• Franklin, P. (1934). "A six color problem". MIT Journal of Mathematics and Physics . 13: 363–379. hdl:2027/mdp.39015019892200.
• Heawood, P. J. (1890). "Map colour theorem". Quarterly Journal of Mathematics . 24: 332–338.
• Ringel, G.; Youngs, J. W. T. (1968). "Solution of the Heawood map-coloring problem". Proceedings of the National Academy of Sciences of the United States of America . 60 (2): 438–445. doi:10.1073/pnas.60.2.438. MR   0228378. PMC  . PMID   16591648.