Grid drawings of k-colourable graphs

Let G = (V,E) be a graph. All graphs considered are simple, £nite and undirected. A grid drawing of G is an injective mapping θ : V → Z such that for all edges vw ∈ E and vertices x ∈ V , θ(x) ∈ θ(v)θ(w) implies that x = v or x = w, where ab denotes the line-segment with endpoints a and b. That is, a grid drawing of a graph represents each vertex by a distinct gridpoint in the plane, and each edge by a line-segment between its endpoints, such that the only vertices an edge intersects are its own endpoints. Let θ be a grid drawing of a graph G = (V,E) such that θ(v) = (X(v), Y (v)) for all vertices v ∈ V . If X(u) − X(v) + 1 ≤ w and Y (u) − Y (v) + 1 ≤ h for all vertices u, v ∈ V , then θ is a w × h grid drawing with area wh and aspect ratio max{w, h}/min{w, h}. This paper studies grid drawings with small area, and with small aspect ratio as a secondary criterion. Minimising the area and aspect ratio are important considerations in graph visualisation for example [2]. Obviously to view a graph drawing with good resolution on a computer screen (which itself has £xed aspect ratio) requires that the area and aspect ratio be small. A k-colouring of a graph G = (V,E) is a partition of V into colour classes V0, V1, . . . , Vk−1 such that for every edge vw ∈ E, if v ∈ Vi and w ∈ Vj then i 6= j. A graph admitting a k-colouring