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 G d as follows:  1)Place a vertex in each region of G; this forms the vertex set of G d.  2)Join two vertices of G d by 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 G d for each bridge that belongs to the corresponding region of G. Moreover, each edge of G d is drawn to cross the associated edge of G, but no other edge of G or G d.

6.  Theorem 5.31 Every 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 P G (k) denote the number of ways of proper coloring the vertices of G with k colors. P G will be called the chromatic function of G.  Example For the graph G P G (k) =k (k-1) 2

10.  If G = (V, E ) with |V | = n and E = , then G consists of n isolated points, and by the product rule P G (k ) = k n.  If G =K n, 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  P G (k ) = k (k-1)(k-2)...(k-n + 1).  We see that for k < n, P G (k ) = 0, which indicates there is no proper k -coloring of K n

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

14.  Suppose that a graph is not connected and G 1 and G 2 are two components of G.  Theorem 5.33: If G is a disconnected graph with G 1,G 2,…G w, then P G (k)=P G 1 (k)P G 2 (k)…P G w (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) OR 324(Fifth) 14,15,26,27  2.In figure 1, find these values  (G),  *(G),  ’(G).   figure 1