Graph theory

A pair of mathematicians solved a legendary question about the proportion of vertices in a graph with an odd number of connections.

Researchers have proved a special case of the Erdős-Hajnal conjecture, which shows what happens in graphs that exclude anything resembling a pentagon.

Fifty years ago, Paul Erdős and two other mathematicians came up with a graph theory problem that they thought they might solve on the spot. A team of mathematicians has finally settled it.

Avi Wigderson and László Lovász won for their work developing complexity theory and graph theory, respectively, and for connecting the two fields.

At 21, Ashwin Sah has produced a body of work that senior mathematicians say is nearly unprecedented for a college student.

David Conlon and Asaf Ferber have raised the lower bound for multicolor “Ramsey numbers,” which quantify how big graphs can get before patterns inevitably emerge.

Two computer scientists found — in the unlikeliest of places — just the idea they needed to make a big leap in graph theory.

Mathematicians have long grappled with the reality that some problems just don’t have solutions.

“Rainbow colorings” recently led to a new proof. It’s not the first time they’ve come in handy.