Hiwood number

The Hiwood number of a surface is a certain upper bound for the maximum number of colors needed to color any graph embedded in a surface.

In 1890, Hywood proved for all surfaces except the sphere that no more than

H(S)=\left\lfloor {\frac {7+{\sqrt {49-24e(S)}}}{2}}\right\rfloor

{\ displaystyle H (S) = \ left \ lfloor {\ frac {7 + {\ sqrt {49-24e (S)}}} {2}} \ right \ rfloor}

{\ displaystyle H (S) = \ left \ lfloor {\ frac {7 + {\ sqrt {49-24e (S)}}} {2}} \ right \ rfloor}

colors necessary for coloring any graph embedded in a surface with an Euler characteristic $e(S)$ ${\ displaystyle e (S)}$ ${\ displaystyle e (S)}$ ^[1] . The case of the sphere corresponds to the four-color hypothesis , which was proved by and in 1976 ^[2] ^[3] . Number $H(S)$ ${\ displaystyle H (S)}$ ${\ displaystyle H (S)}$ became known as the Howwood number in 1976.

Franklin proved that the chromatic number of a graph embedded in a Klein bottle can reach $6$ ${\ displaystyle 6}$ $6$ but never surpasses $6$ ${\ displaystyle 6}$ $6$ ^[4] . It was later proved in the works of Gerhard Ringel and J.W. T. Youngs that a complete graph with $H(S)$ ${\ displaystyle H (S)}$ ${\ displaystyle H (S)}$ vertices can be embedded in the surface $S$ ${\ displaystyle S}$ $S$ , except when $S$ ${\ displaystyle S}$ $S$ is a bottle of Klein ^[5] . This shows that the Hiwood border cannot be improved.

For example, a complete graph with $7$ ${\ displaystyle 7}$ $7$ vertices can be embedded in a torus as follows:

Notes

↑ Heawood, 1890 , p. 322–339.
↑ Appel, Haken, 1977 , p. 429-490.
↑ Appel, Haken, Koch, 1977 , p. 491-567.
↑ Franklin, 1934 , p. 363–379.
↑ Ringel, Youngs, 1968 , p. 438-445.

Literature

Béla Bollobás. Graph Theory: An Introductory Course. - Springer-Verlag, 1979. - T. volume 63. - (GTM). - ISBN 0-387-90399-2 .
Thomas L. Saaty, Paul Chester Kainen. The Four-Color Problem: Assaults and Conquest. - Dover, 1986.

Heawood PJ Map Coloring Theorems // Quarterly J. Math. Oxford Ser .. - 1890. - T. 24 .
Kenneth Appel, Wolfgang Haken. Every Planar Map is Four Colorable. I. Discharging // Illinois Journal of Mathematics. - 1977. - T. 21 , no. 3 .
Kenneth Appel, Wolfgang Haken, John Koch. Every Planar Map is Four Colorable. II. Reducibility // Illinois Journal of Mathematics. - 1977. - T. 21 , no. 3 .
Franklin P. A Six Color Problem // Journal of Mathematics and Physics. - 1934. - T. 13 , no. 1–4 . - DOI : 10.1002 / sapm1934131363 .
Gerhard Ringel, Youngs JWT Solution of the Heawood Map-Coloring Problem // Proceedings of the National Academy of Sciences. - 1968. - T. 60 , No. 2 . - ISSN 0027-8424 . - DOI : 10.1073 / pnas.60.2.438 .

[_ad2fd69910f52d67-1] Heawood, 1890 , p. 322–339.

[_5f1bb1222d68bafb-2] Appel, Haken, 1977 , p. 429-490.

[_890ec97ccf1aeb62-3] Appel, Haken, Koch, 1977 , p. 491-567.

[_91b765b9bca479d9-4] Franklin, 1934 , p. 363–379.

[_7b96924335c1f4f6-5] Ringel, Youngs, 1968 , p. 438-445.

Hiwood number

Notes

Literature

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