Graph colouring is a fundamental problem in both theoretical and applied combinatorics, with significant implications for computer science, operational research and network theory. At its essence, ...

In 1950 Edward Nelson, then a student at the University of Chicago, asked the kind of deceptively simple question that can give mathematicians fits for decades. Imagine, he said, a graph—a collection ...

The Monthly publishes articles, as well as notes and other features, about mathematics and the profession. Its readers span a broad spectrum of mathematical interests, and include professional ...

It is shown that for arbitrary positive integers $m, n$ there exists a $\phi(m, n)$ such that if $\chi(G) \geqslant \phi(m, n)$, then $G$ contains either a complete ...