Sandi Miller | Department of Mathematics at Massachusetts Institute of Technology says, A math professor gives his undergraduates a frustrating combinatorics problem; their solution will soon be published in a leading journal.
In 2009, when Yufei Zhao was an MIT undergraduate, he was intrigued
by a 2001 conjecture by Rutgers University mathematician Jeff Kahn
regarding the number of independent sets in a graph. An independent set
in a graph is a subset of vertices such that no two of them are joined
by an edge.
“Many important structures can be modeled using independent sets,”
said Zhao. “For example, if the graph models some kind of
incompatibility, then an independent set represents a mutually
compatible collection.”
Zhao was participating in a Research Experience for Undergraduates
(REU) summer program in Duluth, Minnesota, and while he was researching
what would be one of his first math research papers, he came across a
combinatorics problem by Kahn. The problem puzzled him. An attempt to
solve it came close, as he described in a paper he wrote with David Galvin in 2010 titled, “The number of independent sets in a graph with small maximum degree.”
Yufei Zhao graduated in 2010, received his PhD in 2015, and is now
the Class of 1956 Career Development Assistant Professor at MIT.
His focus is on combinatorics, discrete mathematics, and graph theory...
The techniques that they found to solve that conjecture quickly led to
work on several related problems, including for their upcoming paper “A Reverse Sidorenko Inequality,”
related to graph colorings and graph homomorphisms.
Explains Zhao: “This paper solves several open problems concerning graph
colorings and homomorphisms, including one of my favorite problems
regarding maximizing the number of q-colorings in a d-regular graph.”
Read more...
Source: MIT News