2. Region(face) colourings

1. 2. Region(face) colourings • Definitions 46: A edge of the graph is called a bridge, if the edge is not in any circuit. A connected planar graph is called a map, If the graph has not any bridge. • Definition 47: A proper region coloring of a map G is an assignment of colors to the region of G, one color to each region, such that adjacent regions receive different colors.An proper region coloring in which k colors are used is a k-region coloring. A map G is k-region colorable if there exists an s-coloring of G for some s  k. The minimum integer k for which G is k- region colorable is called the region chromatic number. We denoted by *(G). If *(G) = k, then G is k-region chromatic.

4. Four Colour Conjecture Every map (plane graph) is 4-region colourable. • Definition 48：Let G be a connected plane graph. Construct a dual Gdas follows: • 1)Place a vertex in each region of G; this forms the vertex set of Gd. • 2)Join two vertices of Gdby an edge for each edge common to the boundaries of the two corresponding regions of G. • 3)Add a loop at a vertex v of Gdfor each bridge that belongs to the corresponding region of G. Moreover, each edge of Gdis drawn to cross the associated edge of G, but no other edge of G or Gd.

6. Theorem 5.31Every planar graph with no loop is 4-colourable if and only if its dual is 4-region colourable.

7. 3. Edge colorings • Definition 49:An proper edge coloring of a graph G is an assignment of colors to the edges of G, one color to each edge, such that adjacent edges receive different colors.An edge coloring in which k colors are used is a k-edge coloring. A graph G is k-edge colorable if there exists an s-edge coloring of G for some s k. The minimum integer k for which G is k-edge colorable is called the edge chromaticumber or the chromatic index ’(G) of G. If ’(G) = k, then G is k-edge chromatic.

9. 4. Chromatic polynomials • Definition 50: Let G =(V, E) be a simple graph. We let PG(k) denote the number of ways of proper coloring the vertices of G with k colors. PG will be called the chromatic function of G. • Example For the graph GPG(k) =k (k-1)2

10. If G = (V, E ) with |V | = n and E =, then G consists of nisolated points, and by the product rule PG(k ) = kn. • If G =Kn, the complete graph on n vertices, then at least n colors must be available for a proper coloring of G. Here, by the product rule • PG(k ) = k (k-1)(k-2)...(k-n + 1). • We see that for k < n, PG(k ) = 0, which indicates there is no proper k -coloring of Kn

11. Let G = (V, E ) be a simple connected graph. For e = {a, b}E, let Ge denote the subgraph of G obtained by deleting e from G, without removing the vertices a and b. Let Ge be the quotient graph of G obtained by merging the end points of e. • Example: Figure below shows the graphs Ge and Ge for the graph G with the edge e as specified.

12. Theorem 5.32 Decomposition Theorem for Chromatic Polynomials (色多项式分解定理) : If G = (V, E) is a connected graph and eE, then • PG(k)=PGe(k)-PGe(k)

14. Suppose that a graph is not connected and G1 and G2 are two components of G. • Theorem 5.33: If G is a disconnected graph with G1,G2,…Gw, then PG(k)=PG1(k)PG2(k)…PGw(k).

15. Chapter 6 Abstract algebra • Groups • Rings • Field • Lattics and Boolean algebra • Next:Abstract algebra, Operations on the set 9.1, P344 (Sixth) OR P330 (Fifth) • Semigroups,monoids and groups 9.2 P349 (Sixth) OR P341 (Fifth),9.4 P362 (Sixth) OR p347 (Fifth)

16. Exercise: P338 (Sixth) OR324(Fifth) 14,15,26,27 • 2.In figure 1, find these values  (G), *(G), ’(G). • figure 1