MT4514: Graph Theory

30 Colva M. Roney-DougalExample 5.10. Consider the following graph G, and let us use the deletion/contractionlemma to find P k (G). We haveSo the number of ways of colouring G isk(k − 1)(k − 2)(k − 1) 2 − k(k − 1)(k − 2) 2 = k(k − 1)(k − 2)[(k − 1) 2 − (k − 2)]= k(k − 1)(k − 2)[k 2 − 2k + 1 − k + 2]= k(k − 1)(k − 2)[k 2 − 3k + 3]Therefore χ(G) = 3.Definition 5.11. A polygon is a planar circuit satisfying v = e.Lemma 5.12. Let G be a graph consisting of the vertices and edges of some polygon,together with a set of non-crossing diagonals of the polygon as further edges.Then G is 3-colourable.Proof. We prove this by induction on the number of vertices. It clearly holdsfor n = 3 vertices:We assume inductively that the result holds for all graphs of this type with lessthan n vertices. Let G be such a graph with n vertices. If G is itself a polygonthen G is clearly 3-colourable, so assume that G contains at least one diagonal v 1 v 2 .Split G into two smaller graphs along v 1 v 2 . Each of the two graphs to either sideof the diagonal satisfies the hypotheses of the lemma, and has less than n vertices.Therefore each part may be 3-coloured by induction. By permuting the colours onone of the parts so that v 1 and v 2 are assigned the same colour on both parts, weget a 3-colouring of G. Example 5.13. Guarding an Art Gallery Let an art gallery be in the shape ofa polygon with n sides. What is the least number of guards that can be placed inthe gallery so that they can see every point in the gallery?The first of these graphs requires at least two guards, the second requires at leastone guard in each shaded region.Proposition 5.14. A polygonal art gallery with n sides may be guarded by ⌊n/3⌋guards.