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, ...

Let p be a positive real number and S(p) the circle in the plane with circumference p. A subset U of S(p) is said to be assignable if it is the union of finitely many open arcs on S(p). Let length(U) ...

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 ...