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Discrete Math

The number of 4-colorings of the Hamming cube

Jinyoung Park, Rutgers University

Location:  Hill 705
Date & time: Monday, 01 October 2018 at 2:00PM - 3:00PM


Let \(Q_d\) be the \(d\)-dimensional Hamming cube (hypercube) and $N=2d$. We discuss the number of proper (vertex) colorings of \(Q_d\) given \(q\) colors. It is easy to see that there are exactly 2 of 2-colorings, but for \(q >2\), the number of \(q\)-colorings of \(Q_d\) is highly nontrivial. Since Galvin (2002) proved that the number of 3-colorings is asymptotically \(6e2^{N/2}\), the other cases remained open so far. In this talk, we prove that the number of 4-colorings of \(Q_d\) is asymptotically \(6e2^N\), as was conjectured by Engbers and Galvin in 2012. The proof uses a combination of information theory (entropy) and isoperimetric ideas originating in work of Sapozhenko in the 1980's.

This is joint work with Jeff Kahn.

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