1 / 16

2. Region(face) colourings

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.

Download Presentation

2. Region(face) colourings

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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.

  2. 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.

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

  4. 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.

  5. 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

  6. 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

  7. 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.

  8. 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)

  9. 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).

  10. 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)

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

More Related