To cement the question, consider a simple example: a graph with three connected vertices in the shape of a triangle. You can isolate any two vertices and see that they share an odd number of ...

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 ...

De Grey's graph has 1,581 vertices on it. And they're arranged in such a way that you couldn't paint it just right with four colors of paint. At least five are necessary to make it work.

Consider a graph on randomly scattered points in an arbitrary space, with any two points x, y connected with probability ϕ (x, y). Suppose the number of points is large but the mean number of isolated ...