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Checking for $3$-colorability by searching for $K_4$ as a minor

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Problem Detail: 

Determining if a graph is $3$-colorable is NP-complete. But this is equivalent, by a resolved case of Hadwiger's Conjecture, to checking if the graph has $K_4$ as a minor.

Per, this can be done in $O(n^3)$ time, where $n$ is the number of vertices, as $K_4$ has a constant number of vertices. This would seem to be a contradiction (unless $P=NP$)

Asked By : vukov
Answered By : Yuval Filmus

Hadwiger's conjecture states that if a graph is not $(k-1)$-colorable then it has a $K_k$ minor. Hadwiger himself proved it for $k \leq 4$. Thus, any graph which is not $3$-colorable has a $K_4$ minor. However, the converse doesn't hold: some graphs having a $K_4$ minor are $3$-colorable. For example, take $K_4$ and split each edge into a path of length $2$ edges. The result is bipartite, and so $2$-colorable.

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