Franklin Graph

 

The Franklin graph is the 12-vertex cubic graph shown above whose embedding on the Klein bottle divides it into regions having a minimal coloring using six colors, thus providing the sole counterexample to the Heawood conjecture. The graph is implemented in the Wolfram Language as GraphData["FranklinGraph"].

It is the 6-crossed prism graph.

The minimal coloring of the Franklin graph is illustrated above.

The Franklin graph is nonplanar but Hamiltonian. It has LCF notations  and .

The graph spectrum of the Franklin graph is .

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REFERENCES

Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 244, 1976.

Franklin, P. "A Six Color Problem." J. Math. Phys. 13, 363-379, 1934.

مواضيع ذات صلة

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Maximum Leaf Number

Lobster Graph

Labeled Tree

Internal Path Length

Red-Black Tree

Pseudotree

Pseudoforest

Prüfer Code

Planted Tree

Planted Planar Tree

Graph Strength