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Hadwiger Conjecture


The Hadwiger conjecture is a generalization of the four-color theorem which states that for any loopless graph G with Hadwiger number h(G) and chromatic number chi(G),

 h(G)>=chi(G)

(Hadwiger 1943).

The case h(G)=5 is equivalent to the four-color theorem, so the proof of the latter established the conjecture for this case. The conjecture was subsequently proven for h(G)=6 by Robertson et al. (1993). However, while the validity of the conjecture has been established for all graphs G with h(G)<=6, it remains open for larger values.


See also

Chromatic Number, Four-Color Theorem, Hadwiger-Nelson Problem, Hadwiger Number

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References

Hadwiger, H. "Über eine klassifikation der Streckenkomplexe." Vierteljschr. Naturforsch. Ges. Zürich 88, No. 2, 133-142, 1943.Robertson, N.; Seymour, P.; and Thomas, R. "Hadwiger's Conjecture for K_6-Free Graphs." Combinatorica 13, 279-361, 1993.Wagner, K. "Über eine Eigenschaft der ebenen Komplexe." Math. Ann. 114, 570-590, 1937.

Cite this as:

Weisstein, Eric W. "Hadwiger Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HadwigerConjecture.html

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